Car Paint Bumper Question?
Your paint will only stick as good as the paint underneath it that you are painting over. Bumpers are notorious for paint peeling off of them because they are difficult to prep. If you paint over some paint that is not adhering well it will cause the old paint to lift off. You should sand all of your paint off and all of the old paint as well. Once you are down to the plastic, you can be sure that you can prepare the surface properly for painting. You need an plastic adhesion promoter spray to start. Also, you need a flex additive for your clearcoat or single stage paint so it doesn't crack. I don't know what your experience or knowledge of paint work is, but there are many things that can go wrong. You can get it professionally painted, which may cost you less money in the long run vs. the cost of continually buying paint and having it not turn out right. If learning how to paint cars is a hobby of yours, then I would encourage you to talk to somebody that works in a shop for some pointers. Car painting can be fun, but if you are wandering around in the dark and just guessing is will probably be more frustrating than fun. Good luck!
Painting requires the area to be prepped properly. You don't state how you prepped the bumper, whether you primed it, and what kind of paint you used. You need to make sure the surface is clean, sanded (with 400 or 600 grit paper), free of oil (don't touch the surface with your hands!), and use a degreaser before you lay down the primer. Use light even coats. Note also that temp and humidity affects the quality of a paint job greatly.
Use a better paint and sand the area with wet sand paper first.