In my experience they don't. They want to know if you're a reasonable, self-aware person who can communicate effectively.
Let's say you name a reasonable weakness that can easily be improved upon and that doesn't spell disaster for your ability to do your job. And let's say they reject you.
Either they're a terrible company to work for, or they have a huge mistake in their interview process. In the former case you don't want to work for a terrible company, so great, they just saved you years of pain. In the latter case, hey, mistakes happen. That's out of your control. Life is random sometimes. Besides, it's highly unlikely to be significant enough to tank your entire interview.