I assume you're allowing A to be non-commutative.  In this case, things can go wrong in all kinds of ways.  For example, all Verma modules have endomorphisms given by \CC, but loads on those are not simple.  Actually, every standard object in every highest weight category has endomorphisms given by a divison algebra.

Another good example is that if you look at the path algebra of a Dynkin diagram, all indecomposible modules have endomorphisms given by the base field.  By Gabriel's theorem, simples are in bijection with simple roots, indecomposibles are in bijection with positive roots, so there are lots of these.