The material that these cabinets is made out of is called melamine. It is basically an inexpensive particle board that is covered with some sort of plastic coating, I think it is vinyl. I don't have much experience with painting it but I would wager that it could be ideal for painting IF you prep the surface correctly. What you MUST do is: wash, sand, prime, paint.
Buy a strong pre paint cleaner that will leave no residue. There must be no hint of grease, dirt or wax or it might not work out. Next you must sand every surface that will be painted. I would use a 150 grit on a sanding block. Just a few passes are needed, don't wear through the vinyl, just scuff it up. Be very thorough because if you miss a spot, the paint won't stick. I would use Kiltz (sp.?) water based super duper primer before the finish paint. I would pull off all of the doors and hinges and hardware and spray them with an air sprayer for the best results.
Hey Margaret Fisher, did you follow all of these steps or did you not rough up the surface first? - just curious... jb