We've all done it: we have a "proof" that is so pretty... but we know it's wrong. Either it proves something we know is false, or the proof doesn't use one of the hypotheses you know it needs to, or we assume something we can't assume... and the whole pretty argument goes away. I had a professor who said that he sometimes convinces himself that some proof works right before he goes to bed, only to wake up and find out that it's total nonsense. He says it's probably some coping mechanism his body has so he can get some sleep!
Though the proofs often end up unsalvageable (is that a word?), I think they are an important part of learning mathematics and being creative. I think it was Sir Ken Robinson (not a mathematician, but a good thinker) who said something like "Being wrong isn't always good, but if we don't have the capacity to be wrong we can never be creative." (If anyone knows the precise quote, do fill it in!)
So let's hear them- your pretty proofs that turn out to be nonsense. Like the well-written "How not to prove the Poincare Conjecture," these can be well thought out proofs that have a small hidden assumption that makes everything blow-up, or they can be totally silly like the time I thought I proved Brouwer's fixed point theorem using the Baire Category Theorem during my freshman year...
Bonus points if anyone's "false" methods (or a close adaptation) ended up working for a different problem later on!