If you give the police information that turns out to be inaccurate, and the police mistakenly believe that you were lying to them on purpose, that fact can be devastating to your defense in three different ways. First, it can help to convince the police that they have the right suspect, which might make them less likely to spend additional time pursuing other possible leads that could help them identify the actual offender. Second, the prosecutor can present that evidence to the jury, and the judge will tell the jurors that, if they believe that you knew your statement to the police was false when you said it, they are permitted to regard that knowing falsehood as evidence that you are guilty. (And how will the jury ever really know whether you are lying to the police? They are only human, just like the cop.) Third, and perhaps worst of all, our legal system places no limits on the ability of the police to share the details of their ongoing investigation with the critical witnesses against you.
James Duane in You Have the Right to Remain Innocent