What is the Natural Size of Supersymmetric $CP$ Violation?

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 $10^{-3}$.