Sir Marbelot - Yes, exactly.
Larry - Yeah, they are brittle (hardness = brittle).

Quite often I'll have to dress up a broken screw with the dremel to get a suitable landing area for the drill bit. And then I bisect the target area with a small cold chisel and then use a center punch to mark where I'll start with a small drill bit. It sounds like a PITA, but it's really not and it makes life much easier to start off the hole exactly where you want it instead of 'close'.
I'll usually try a screw extractor first before resorting to drilling out the entire screw. If I'm successful at getting the screw out, then I'll run a tap through it all the same just to clean the threads. If I end up drilling it out, then I'll go oversize. Quite often the old screw body will pop out when you're stepping up through the drill sizes and there's not much material left in the threads. I used to have this reverse pitch drill bit set that was great at torquing out broken screws, but they would break quite often itself and I never replaced them.