  We suggest that if a localized phase at nonzero temperature $T>0$ exists for
strongly disordered and weakly interacting electrons, as recently argued, it
will also occur when both disorder and interactions are strong and $T$ is very
high. We show that in this high-$T$ regime the localization transition may be
studied numerically through exact diagonalization of small systems. We obtain
spectra for one-dimensional lattice models of interacting spinless fermions in
a random potential. As expected, the spectral statistics of finite-size samples
cross over from those of orthogonal random matrices in the diffusive regime at
weak random potential to Poisson statistics in the localized regime at strong
randomness. However, these data show deviations from simple one-parameter
finite-size scaling: the apparent mobility edge ``drifts'' as the system's size
is increased. Based on spectral statistics alone, we have thus been unable to
make a strong numerical case for the presence of a many-body localized phase at
nonzero $T$.
