  Extremal charged black holes are BPS solutions. It is commonly thought that
their nonextremal counterparts are not. Further, experience with BPS solutions
in flat spacetime suggests that all BPS solutions are supersymmetric; i.e. that
they are invariant under some supersymmetry charges of either the original
field theory or an appropriately extended version thereof. Using nonextremal
Reissner-Nordstr\"om black holes as counterexamples, we show that neither of
these expectations is universally valid. These black holes correspond to a
one-parameter family of BPS solutions. By showing that, subject to one very
plausible assumption, no generalized Killing spinor can be constructed for
these, we show that there is no supergravity theory for which these BPS
solutions preserve a fraction of the supersymmetry, nor is there an associated
Witten-Nester positive energy bound.
