Anecdotal Evidence

If you discount evidence arrived at by systematic search or by testing in favor of a few firsthand stories, you are committing the fallacy of overemphasizing anecdotal evidence.

Example:

Yeah, I've read the health warnings on those cigarette packs and I know about all that health research, but my brother smokes, and he says he's never been sick a day in his life, so I know smoking can't really hurt you.