prosecutor's fallacy
http://en.wikipedia.org/wiki/Prosecutor%27s_fallacy
It is precisely the failure to apply Bayes' theorem that is the most glaring abuse of statistical "reasoning" in most trials. But applying it requires lots of numbers, and it you estimate them it is far too easy to put a thumb on the scale, subconsciously or otherwise.
the point made by "xlq" is also a hot topic in statistical circles. One aspect of it is the Birthday paradox: if you have a large enough population, coincidences will happen.
A related problem does not have such a catchy name. If you do a whole lot of tests for statistically significant relationships among many variables that are actually not related, you will get some that test as significant at random. See recent articles about the problem of observational studies in health and medicine.
In criminal investigation, it is one thing to get a "five point match" on a fingerprint AFTER identifying a likely suspect, it is something else entirely to have a computer spit out a five-point match from a database of millions of people's prints.