LEVAL, Circuit Judge:

Jeffrey Stein was convicted of conspiracy to import more than 5 kilograms of cocaine after a jury trial before Warren W. Eginton, Judge, in the District of Connecticut, and was sentenced to 135 months imprisonment, and other penalties. We reverse because of prejudicial improper evidence that the Government placed before the jury.

The prosecution arose out of an investigation conducted by the United States Customs Service after receiving a tip that the cargo ship Potomac, arriving in Bridgeport harbor, would have cocaine secured to the hull. Upon the ship's arrival, Customs supervised an underwater search by scuba divers who discovered a tube attached to the ship's keel. The tube was found to contain 66 kilograms of cocaine. Expecting drug traffickers to attempt to remove the cocaine during the ship's brief stop in port, Customs established a surveillance and investigation, which led the agents first to Francisco, Fernando and Berta Reyes, and from them to Rafael Reyes and the defendant Jeffrey Stein.

The indictment charged Stein together with the four members of the Reyes family. Before trial Fernando, Francisco and Rafael Reyes pleaded guilty, and the charges against Berta were dismissed. Stein accordingly stood trial alone.

The principal witness against Stein was Rafael Reyes, who had agreed to cooperate with the Government. Rafael thoroughly implicated Stein in the plot.

The Government offered corroborative evidence, partly through the testimony of Customs Agent Maryann Caggiano. Agent Caggiano's testimony was presented in the form of a narration of the exciting story of the investigation. Starting with Customs' receipt of the tip about narcotics affixed to the hull of the Potomac, she testified that Customs set up a scuba operation in Bridgeport harbor and discovered the tube affixed to the keel containing 66 kilos of cocaine; she established surveillance of the port, and received a tip from a tavern owner giving the license plates of a suspicious red van; she traced the license to a Stamford car rental agency and learned that the van had been rented to a Francisco Reyes; she watched the car rental agency awaiting the return of the van and confronted Fernando and Francisco Reyes following its return; she interviewed them, searched the trunk of their other car with their consent, finding scuba gear and tools appropriate for the removal of the cocaine canister from the hull of the ship; and arrested them. Caggiano went on to testify that the investigation led from those three members of the Reyes family to Rafael Reyes in Chicago and to the defendant Stein in Queens. From there, she described the early phases of the prosecution — the grand jury's return of an indictment, the issuance of a warrant to arrest Stein and proceedings on a motion to suppress evidence. (While much of Caggiano's testimony was relevant to the scope of the conspiracy and its membership, the narrative included many facts that were irrelevant to the question of Stein's guilt, including testimony concerning her own state of mind as the investigation progressed and the early events in the institution of the prosecution.)

During Caggiano's testimony (and the prosecutor's summation references to it), there occurred episodes which, in our view, require reversal of Stein's conviction.

A. Fernando and Francisco implicate Stein. During Caggiano's narration of her initial confrontations of Francisco and Fernando, following the return of the red van, the following testimony was given:

Q. Now, did you have further discussions with [Fernando and Francisco] at sometime after one o'clock on September 20 of 1990?

A. Yes.

Q. And did those further discussions with these individuals cause you to believe that there were other people involved with them in this particular criminal activity?

A. Yes, I did.

Q. And who were those two individuals?

. . . . .

A. Would you repeat the question?

. . . . .

Q. Yes. As a result of your further conversations, did you come to a conclusion that there were other individuals involved in this criminal enterprise?

A. Yes, I did.

Q. And who were those other individuals?

A. Rafael Reyes and Jeffrey Stein.

Then after testifying about the discovery of the scuba gear and the tools in the Reyeses' car, Caggiano testified as follows:

Q. After you made those observations, did you have any further discussions with these two individuals?

A. Yes.

Q. And as a result of those discussions, did you learn whether Mr. Stein had been in Connecticut?

A. Yes, I did.

In both cases, this testimony was received over the defendant's hearsay objection and provoked a motion for a mistrial. In answer the Government successfully argued that the evidence should be received not for the truth of the matter asserted but as background, showing the agent's state of mind and the reasons for her actions.

B. Fernando describes the matchbook. After narrating her arrest of Fernando and Francisco, Caggiano was asked whether Fernando "directed [her] to a particular document that he thought was significant." The agent said "yes" and referred to a matchbook cover, later introduced as GX 23-A. The Assistant United States Attorney asked, "Ms. Caggiano, what was the significance of that matchbook?" to which the agent answered, "Mr. reyes had stated that the numbers on the matchbook cover were beeper numbers for two people in Colombia that he was to get in contact with." Also written in the matchbook was Stein's address in Queens.

Upon summation, referring to the matchbook cover, the Assistant said:

Special Agent Caggiano told you that while they were in the — questioning Mr. Francisco and Fernando Reyes, there was a statement made about a matchbook cover with numbers of individuals who were responsible for the cocaine. . . .

If you look at the matchbook cover that the Reyeses said contained the information about the people responsible for this load of cocaine, what appears on it? You have the name Herman. You have the name Bolero and Carlos, telephone numbers next to it, and on the bottom of the matchbook cover you've got once again 1418 31st Drive, Astoria, Queens. Once again, Mr. Stein's address and the very piece of paper they pulled out being important to the owners of this cocaine, the people responsible for this cocaine. (emphasis added).

Defense counsel objected to this use of Fernando's out-of-court statement for the truth of what he had said and further to the misrepresentation of Caggiano's testimony as to what he had said.

As to the objection to the use of hearsay, the Assistant responded, "There was no objection at the time [of the receipt of the evidence], so any objection is waived." As to the accuracy of his representation of the testimony, the Assistant said, "My recollection is that indeed was in the evidence, and I commented upon — over that which was in the evidence. That's my recollection." The objection was overruled and the motion for mistrial denied.

C. The prejudicial question asked and withdrawn. A third episode narrowly averted an additional reason to reverse the conviction:

After her narration of the arrest of Fernando and Francisco Reyes, Caggiano was asked whether she had conducted another interview of Fernando Reyes, which she confirmed.

Q. Did the results of that interview confirm your earlier suspicions about the other co-conspirators in this conspiracy venture?

The defendant again objected on the grounds of hearsay. The Assistant responded:

The Government's position is that it's perfectly appropriate. As both Mr. Ward and the court obviously knows, I haven't made any attempt to elicit the substance of those conversations. The reason the Government is eliciting it is it shows what effect that had on the mind of Ms. Caggiano.

. . . . .

However, that being said, the Government is going to withdraw the question, despite the fact I think it's perfectly appropriate under the rules. . . .

Had the witness answered the leading question, this would have compounded the prejudice by telling the jurors that Fernando had again confirmed Stein's complicity in the conspiracy.

Discussion

I. Hearsay

Hearsay is evidence of a declarant's out-of-court statement to prove the truth of what is asserted in the statement. See Fed. R.Evid. 801; 5 Wigmore, On Evidence § 1364 (3d ed. 1974); 2 McCormick, On Evidence § 246 (4th ed. 1992). The principal vice of hearsay evidence is that it offers the opponent no opportunity to cross examine the declarant on the statement that establishes the declared fact.

The government contends that the passages quoted under A involved no hearsay because (i) the declarant's words were not repeated and (ii) in any event the jury was instructed that the Reyeses' statements were received not for their truth but only to explain Agent Caggiano's state of mind.

We reject the first contention: although the jury was not told exactly what words Francisco and Fernando had spoken, Caggiano's testimony clearly conveyed the substance of what they had said. The clear message conveyed to the jury (especially when connected to the hearsay about the numbers on the matchbook) was that Francisco and Fernando had admitted their role in a conspiracy to import the cocaine and had implicated Rafael and the defendant Jeffrey Stein. It also effectively communicated their assertion that Stein (who lived in Queens) had been in Connecticut with them.

As to the government's second argument, it is technically correct. Because the jury was instructed not to consider the out-of-court declarations as proof of the truth of what was said, technically no hearsay evidence was received. However, when the likelihood is sufficiently high that the jury will not follow the limiting instructions, but will treat the evidence as proof of the truth of the declaration, the evidence is functionally indistinguishable from hearsay. Whether such evidence may be received turns not merely on the delivery of a limiting instruction that the jury may be unable to follow, but on a more complex balancing of factors that is discussed below.

In the episode involving the matchbook cover ( B above), Caggiano explicitly repeated Fernando's words, telling that Fernando had said the matchbook contained the "beeper numbers for two people in Colombia that he was to get in contact with." The defense attorney, whose hearsay objections had already been rejected repeatedly upon the Government's representation that the Reyeses' statements were offered not for their truth but for background and the agent's state of mind, did not renew the objection.

In addition, the account given to the jury in the summation differed significantly from what Fernando was reported by Caggiano to have said. Caggiano's testimony of Fernando's statement was that the matchbook showed the "beeper numbers for two people in Colombia that he was to get in contact with." That statement clearly could not have referred to Stein's address in Queens. However, when the Assistant referred to this testimony in his summation, he told the jury that "the Reyeses said [the matchbook cover] contained the information about the people responsible for this load of cocaine" and then pointed to Stein's address on the matchbook cover. Thus, whereas Fernando had in fact spoken only of the beeper numbers of the Colombians, the account reported to the jury in the summation sounded as if the Reyeses had explicitly identified Stein as one of the "people responsible for the cocaine."

We are assured by the Government and are fully convinced that the discrepancy between Caggiano's testimony and the summation was not intentional. Although the mistake had innocent origins, our concern is for its possible effect on the jury, especially in that it was coupled with the other hearsay testimony that communicated Fernando's implication of Stein.

II. The Balance of Probative Value Against Prejudice

It is true, as the Government contends, that, in some instances, information possessed by investigating agents is received at trial not for the truth of the matter, but as "background" to explain the investigation, or to show an agent's state of mind so that the jury will understand the reasons for the agent's subsequent actions. See, e.g., United States v. Lubrano, 529 F.2d 633, 637 (2d Cir. 1975), cert. denied, 429 U.S. 818, 97 S.Ct. 61, 50 L.Ed.2d 78 (1976); United States v. Valencia, 957 F.2d 1189, 1198 (5th Cir.), cert. denied, ___ U.S. ___, 113 S.Ct. 254, 121 L.Ed.2d 185 (1992); United States v. Love, 767 F.2d 1052, 1063 (4th Cir. 1985), cert. denied, 474 U.S. 1081, 106 S.Ct. 848, 88 L.Ed.2d 890 (1986). Such evidence can be helpful in clarifying noncontroversial matter without causing unfair prejudice on significant disputed matters. In other instances, it can constitute appropriate rebuttal to initiatives launched by the defendant. See United States v. Hawkins, 905 F.2d 1489, 1495 (11th Cir. 1990), cert. denied, 498 U.S. 1038, 111 S.Ct. 707, 112 L.Ed.2d 696 (1991). However, neither circumstance obtained here. At trial and on this appeal, the Government has displayed so egregious a misunderstanding of the circumstances that will justify such evidence that the subject requires some explanation.

The proffer of such evidence generally raises two questions: first, whether the non-hearsay purpose by which the evidence is sought to be justified is relevant, i.e., whether it supports or diminishes the likelihood of any fact "that is of consequence to the determination of the action," see Fed. R.Evid. 401, and second, whether the probative value of this evidence for its non-hearsay purpose is outweighed by the danger of unfair prejudice resulting from the impermissible hearsay use of the declarant's statement. See Fed.R.Evid. 403. See also United States v. Tussa, 816 F.2d 58 (2d Cir. 1987); United States v. Martin, 897 F.2d 1368 (6th Cir. 1990); United States v. Mancillas, 580 F.2d 1301, 1309 (7th Cir.), cert. denied, 439 U.S. 958, 99 S.Ct. 361, 58 L.Ed.2d 351 (1978). Thus, contrary to the government's contention, the mere identification of a relevant non-hearsay use of such evidence is insufficient to justify its admission if the jury is likely to consider the statement for the truth of what was stated with significant resultant prejudice. The greater the likelihood of prejudice resulting from the jury's misuse of the statement, the greater the justification needed to introduce the "background" evidence for its non-hearsay uses.

Questions involved in the determination of the relevance and importance of such evidence include: (i) Does the background or state of mind evidence contribute to the proof of the defendant's guilt? (ii) If so, how important is it to the jury's understanding of the issues? (iii) Can the needed explanation of background or state of mind be adequately communicated by other less prejudicial evidence or by instructions? (iv) Has the defendant engaged in a tactic that justifiably opens the door to such evidence to avoid prejudice to the Government?

Questions involved in the assessment of potential prejudice include: (v) Does the declaration address an important disputed issue in the trial? Is the same information shown by other uncontested evidence? (vi) Was the statement made by a knowledgeable declarant so that it is likely to be credited by the jury? (vii) Will the declarant testify at trial, thus rendering him available for cross-examination? If so, will he testify to the same effect as the out-of-court statement? Is the out-of-court statement admissible in any event as a prior consistent, or inconsistent, statement? (viii) Can curative or limiting instructions effectively protect against misuse or prejudice?

In this case, virtually every variable argues against receipt of this evidence.

First, as to the relevance and importance of the asserted non-hearsay purpose: Agent Caggiano's state of mind as her investigation progressed was probably not relevant and was certainly not important to any issue properly in the trial. Her state of mind may be relevant to the history of the investigation, but the history of the investigation was not relevant to the guilt or innocence of the defendant. Prosecutors sometimes adopt a tactic of structuring the evidence in the form of the history of the investigation, because it makes the evidence more exciting and perhaps also because it suggests a guilty verdict as a logical, satisfying conclusion. It is not improper for prosecutors to use this tactic in presenting relevant evidence. But the fact that the prosecutor chooses this narrative device does not enlarge the scope of relevant evidence. The investigating agent's state of mind, although perhaps important to the story of the investigation, nonetheless remains irrelevant to the issue being tried: The question of the defendant's guilt.

Even if there had been sufficient reason to explain to the jury why the agent investigated Stein, that explanation was amply provided by the fact that his address had been used by the Reyeses in renting the red van and appeared again on Fernando's matchbook cover.

Nor had the defendant engaged in tactics that would justify rebuttal through the state of mind evidence, such as imputing bias, or likelihood of fabrication of evidence, to the Government agent. Where this happens, it can be important for the Government to show the jury that the agent had a valid reason to investigate the defendant. Such tactics by the defendant can result in opening the door to the receipt of even prejudicial declarations for the non-hearsay purpose of explaining the agent's actions. The Government makes no contention that any tactic of Stein called for such response at trial.

As to the capacity for prejudice: The co-conspirators' statements addressed the most important disputed issue in the trial; they directly implicated the defendant in the crime. In addition, the declaration implicating the defendant in the crime came from a highly credible source. Caggiano's testimony conveyed that Fernando and Francisco admitted their complicity in the cocaine conspiracy, and immediately implicated their relative Rafael and the defendant Stein. Under the circumstances, the hearsay declaration communicated to the jury a powerful message that the defendant was guilty.

Fernando and Francisco did not testify at the trial. Accordingly, the defendant had no opportunity to discredit their declarations to Agent Caggiano by cross-examination.

Ordinarily we assume that juries will follow limiting and curative instructions. There are, however, occasions where the prejudice is so severe that such instructions are unlikely to be effective. Given the high potency of these declarations, their determinative significance for the only important issue in the trial, and their lack of significance for any other purpose, the limiting instructions given by the court were unlikely to prevent the jury from considering the declarations for their truth. The instructions given, furthermore, did not clearly explain the difficult mental task of considering information for one purpose but not for another. In our view, they did not effectively guard against the prejudicial impact of the declarations as highly persuasive evidence of the defendant's guilt. No limiting instruction was given, furthermore, with respect to the matchbook evidence.

In conclusion, the resulting prejudice from the receipt of such incriminating declarations was considerable and far exceeded the minimal or non-existent probative value of the non-hearsay uses of this evidence. We conclude that the receipt of this evidence was error.

Whether these errors should be considered harmless is a closer question. The evidence against Stein was unquestionably strong, but was not overwhelming. And even to the extent the evidence tended to show Stein's involvement in the planned diving operation, this did not necessarily show his knowledge that the dive was for cocaine. The evidence against Stein came primarily from Rafael Reyes, whose credibility was undermined by his motive to obtain a reduction of sentence through cooperation. The Government contends he was rehabilitated by the fact that he had made a pre-arrest statement implicating Stein to a police officer in Chicago. This attempted rehabilitation, however, suffered from two flaws. First, the only evidence that Rafael named Stein in his pre-arrest statement to the Chicago police officer came through Rafael's testimony. If the jury doubted Rafael's truthfulness in implicating Stein, it could equally doubt the veracity of his testimony that he had told the Chicago policeman about Stein. Secondly, even if Rafael made a prior consistent statement to a Chicago policeman before his arrest, that was not necessarily prior to his motive to fabricate. At the time, Rafael knew that Francisco and Fernando had been arrested and that he would probably soon be arrested; the reason he spoke to the Chicago police officer was to explore the possible benefits of cooperation; thus the motive to provide evidence against Stein already existed.

We cannot conclude that it was "`highly probable' that the[se] error[s] did not contribute to the verdict," Tussa, 816 F.2d at 67 (citations omitted), or that these errors had a "very slight effect" on the jury. United States v. Zackson, 12 F.3d 1178, 1184 (2d Cir. 1993). These errors do not pass the harmless error test. The judgment of conviction must be reversed. Because the case will likely be retried, we note for guidance on retrial that we found no merit to Stein's other contentions.

We add a note of caution to criminal prosecutors. Because in criminal cases there has been little prior discovery, and the defense lawyers often do not know in advance what will be the testimony of prosecution witnesses, trial judges have little ability to prevent error if prosecutors act without due caution. The need for retrial of this case could easily have been avoided if the Assistant U.S. Attorney, recognizing that he was about to elicit potentially incendiary evidence as to which there are arguable grounds for exclusion, had begun by a proffer, preferably in writing, explaining the issues in full, so that the defendant had the chance to object and the judge to rule before the harm was done.

Conclusion

The judgment of conviction is reversed and the case remanded to the district court.