Help peeling painted cabinets
To get something that lasts on high traffic touch areas like cabinets you need to go with oil based paint.  The latex stuff just doesn't hold up and it will come off or get rubbery with water, changes in temperature (near oven), or a lot of touching (oils and rubbing it).  This is especially true when latex is on oil without being primed right.

If the cabinet doors are nice I would try to strip them.  A more practical solution is buying an oil based primer made to go over latex.  And then an oil based paint on top of that.  Do your worst offending door.  See how the results are after a few weeks or a couple of months and then do the rest.