A couple of the guys have told you the right answer.
Very thin drill bits don't have much material there... remember, the diameter of the bit is one thing, there's actually much less material there because of the spiral cut.
The trick, as stated, is to chuck the bit so that only as much of the bit that you need to get through whatever you're drilling is showing. The shorter the exposed part of the bit, the better. And let the bit do the work... it takes very little pressure to snap those small bits. I know, because I've done it more times than I want to admit to!