  We continue our analysis of the number partitioning problem with $n$ weights
chosen i.i.d. from some fixed probability distribution with density $\rho$. In
Part I of this work, we established the so-called local REM conjecture of
Bauke, Franz and Mertens. Namely, we showed that, as $n \to \infty$, the
suitably rescaled energy spectrum above some {\it fixed} scale $\alpha$ tends
to a Poisson process with density one, and the partitions corresponding to
these energies become asymptotically uncorrelated. In this part, we analyze the
number partitioning problem for energy scales $\alpha_n$ that grow with $n$,
and show that the local REM conjecture holds as long as $n^{-1/4}\alpha_n \to
0$, and fails if $\alpha_n$ grows like $\kappa n^{1/4}$ with $\kappa>0$.
  We also consider the SK-spin glass model, and show that it has an analogous
threshold: the local REM conjecture holds for energies of order $o(n)$, and
fails if the energies grow like $\kappa n$ with $\kappa >0$.
