Twelve Angry Men, the classic movie discussed in David's post was a great film. But I've always thought that the defendant Henry Fonda's character persuades the jury to acquit was actually guilty. There were four or five separate pieces of evidence pointing to the defendant's guilt, including two separate eyewitnesses. Fonda's character does a good job of showing that no one piece of evidence was enough to prove the accused guilty beyond a reasonable doubt by itself. But he and the other jurors ignore the possibility that guilt might be established by the cumulative weight of multiple pieces of evidence that are individually insufficient. For example, let's assume that the defendant has five pieces of evidence against him and each of them individually shows that there is only a 70% chance of his being guilty. The combined probability of his guilt based on all five items is about 99.8%, more than enough to prove guilt beyond a reasonable doubt. By focusing on each piece of evidence individually, Fonda's character obscures this fact and persuades the jury to let a guilty man go. Obviously, this interpretation of the movie is not the one that the filmmakers wanted the audience to come away with. But I think it fits the evidence nonetheless.
In real life, this "divide and conquer" strategy was effectively used by O.J. Simpson's defense lawyers, who raised doubts about some of the individual items of evidence against their client, but successfully avoided confronting the fact that he was almost certainly guilty based on the cumulative weight of many different items of evidence. Former prosecutor Vincent Bugliosi discusses this in his excellent book on the Simpson case.
The point is applicable to issues beyond criminal law. People often dismiss individual arguments and evidence against their preferred position without considering the cumulative weight of the other side's points. It's a very easy fallacy to fall into. But the beginning of wisdom is to at least be aware of the problem.
UPDATE: My analysis assumes that the five pieces of evidence were conditionally independent of each other (i.e. - that the discrediting of one does not affect the odds of the others being valid). That, I think, is an accurate representation of the evidence in the movie, which consistent of several independent items: two separate eyewitness accounts, some items of physical evidence, and flaws in the defendant's alibi.