In the United States, the Double Jeopardy Clause of the U.S. constitution has been held to prohibit retrying a person for a crime they were previously acquitted of, even if the acquittal was merely due to an insufficiency of evidence. To quote Wikipedia on the subject:
"Retrial is not possible if the verdict is overturned on the grounds of evidentiary insufficiency, rather than on the grounds of procedural faults."
The problem with this is, it is very easy to think of circumstances in which some, or even much, incriminating evidence might not come to light until after the trial (and possible acquittal). One big category would be evidence only made useable by scientific and\or technological advances that hadn't yet happened at the time of the trial; for instance, DNA profiling, which could easily provide damning evidence against a suspect, was not available until the mid-1980s, meaning that there could potentially be any number of criminals running around from before then who couldn't be convicted at the time because evidence that would have been damning, had it been useable at the time, wasn't useable yet, because the techniques for using it simply didn't exist yet.
So why does double jeopardy prevent someone from being tried again, even if new incriminating evidence is found after the fact? It would seem to me that the discovery of new evidence at a later date would serve as proof that the defendant wasn't fully in jeopardy in the earlier trial, since it would show that the prosecution was handicapped by not being able to use evidence that existed, but which had not yet come to light...