How big a hole in drywall can be fixed by spackle alone?
I don't think you'll be happy with spackle for anything bigger across than a nail hole.  It's not very strong, and also not very sticky.  So what tends to happen in your situation is that the spackle will fall through the hole into the wall; if not when you're applying it then it will be likely to do it when you try to sand it smooth before you paint.

I think the solution that's going to get you the best results for holes that size would be to cut a 2" square of fiberglass drywall joint tape and stick it over the hole, and then use joint compound to fill the hole and cover the tape, feathering it out to a 6-8" diameter circle.  You can do this reasonably well with a cheap 4" putty knife.  Then sand it smooth, being careful not to sand all the way through to the tape.