I do use 3rd-party libraries. I'm a big fan of JQuery, for example - I can't use it in my day job because we literally count bytes when serving JavaScript, but I use it all the time for prototypes and personal projects. But I know that Resig is a top-notch coder, and more importantly, JQuery has had thousands of projects beat on it for a couple of years. If there're corner cases, they've been explored and uncovered, bug reports have been filed, and patches have been made.
Similarly, I use Apache/Lighttpd and MySQL and Django and PIL and BeautifulSoup (though I had some installation issues with PIL, so maybe that's not a great example of a well-tested, well-documented library). But these are pretty well-exercised projects with large userbases: other people have already done the dirty work of discovering all the bugs.
Makes me wonder why anyone would ever use new open source projects, which is essential to getting to the well-tested, well-exercised stage. I think it comes down to need: they'll reach for an imperfect library if the problem is so scary that they don't even want to approach it. So go work on hard problems, and you'll be popular. ;-)