  The supersymmetric $SU(3)\times SU(2)\times U(1)$ theory with minimal
particle content and general soft supersymmetry breaking terms has 110 physical
parameters in its flavor sector: 30 masses, 39 real mixing angles and 41
phases. The absence of an experimental indication for the plethora of new
parameters places severe constraints on theories posessing Planck or GUT-mass
particles and suggests that theories of flavor conflict with naturalness. We
illustrate the problem by studying the processes $\mu \rightarrow e + \gamma$
and $K^0 - \bar{K}^0$ mixing which are very sensitive probes of Planckian
physics: a single Planck mass particle coupled to the electron or the muon with
a Yukawa coupling comparable to the gauge coupling typically leads to a rate
for $\mu \rightarrow e + \gamma$ exceeding the present experimental limits. A
possible solution is that the messengers which transmit supersymmetry breaking
to the ordinary particles are much lighter than $M_{\rm Planck}$.
