Another thing to watch for is whether the tap is perfectly in-line with the hole. If it goes in at the slightest angle, it can bind at some point and break. A larger tap can handle more stress, and might be able to cut it's way through if it's off just a tiny bit, but a small one won't have a chance. Also, smaller taps have less room for chips, so backing it out a couple of times to clean it off and blowing out the hole, might help.
The best approach would be to go with the drill press suggestion. Lock the vice down, and the part in it. Drill the hole, remove the bit from the chuck, chuck up the tap, and with the power off, turn the chuck by hand while putting light downward pressure on the handle. This should help keep everything lined up.