  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 the possibility
of finding a gravity/field theory realization of the holographic principle for
a broad class of solutions, including asymptotically flat black holes. 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.
