Is it possible to make an area where you have a composite filling sore and bleed from brushing too hard?

Yes: Some people use their toothbrush like a saw and some use it like feather duster. If the composite is near tissue, of course overbrushing can cut the tissues around the filling which is not a good thing. Lighter brushing and a softer toothbrush can still get the area clean. A good dental hygienist can coach you as to the best way to clean those composites and teeth around your gums.

Yes: You can make any area sore if you brush too hard. Brush gently and thoroughly, but don't overdo it.