We discuss the entropy change due to fragmentation for black hole solutions in various dimensions. We find three different types of behavior. The entropy may decrease, increase, or have a mixed behavior, characterized by the presence of a threshold mass. For two-dimensional ~2D! black holes we give a complete characterization of the entropy behavior under fragmentation, in the form of sufficient conditions imposed on the function J, which defines the 2D gravitational model. We compare the behavior of the gravitational solutions with that of free field theories in d dimensions. This excludes for a broad class of solutions, including asymptotically flat black holes, the possibility of finding a duality between gravity and a field theory, which realizes the holographic principle. We find that the most natural candidates for holographic duals of the black hole solutions with mixed behavior are field theories with a mass gap. We also discuss the possibility of formulating entropy bounds that make reference only to the energy of a system.