  It is well known that if phases and masses in the Minimal Supersymmetric
Standard Model (MSSM) are allowed to have general values, the resulting neutron
EDM ($d_n$) exceeds the experimental upper limit by about $10^3$. We assume
that the needed suppression is not due to a fine-tuning of phases or masses,
and ask what natural size of $CP$ violation (CPV) results. We show that (1) the
phase of one of the superpotential parameters, $\mu$, does not contribute to
any CPV in the MSSM and so is not constrained by \dn; (2) the MSSM contribution
to $d_n$ is tiny, just coming from the CKM phase; (3) the phases in the MSSM
cannot be used to generate a baryon asymmetry at the weak scale, given our
assumptions; and (4) in non-minimal SUSY models, an effective phase can enter
at one loop giving $d_n \sim 10^{-26}$\ecm, $d_e \sim 10^{-27}$\ecm, and
allowing a baryon asymmetry to be generated at the weak scale, without
fine-tunings. Our results could be evaded by a SUSY breaking mechanism which
produced phases for the SUSY breaking parameters that somehow were naturally of
order $10^{-3}$.
