  The existence of a neutrino magnetic moment implies contributions to the
neutrino mass via radiative corrections. We derive model-independent
"naturalness" upper bounds on the magnetic moments of Dirac neutrinos,
generated by physics above the electroweak scale. The neutrino mass receives a
contribution from higher order operators, which are renormalized by operators
responsible for the neutrino magnetic moment. This contribution can be
calculated in a model independent way. In the absence of fine-tuning, we find
that current neutrino mass limits imply that $|\mu_\nu| < 10^{-14}$ Bohr
magnetons. This bound is several orders of magnitude stronger than those
obtained from solar and reactor neutrino data and astrophysical observations.
