  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 and Majorana
neutrinos, generated by physics above the electroweak scale. For Dirac
neutrinos, the bound is several orders of magnitude more stringent than present
experimental limits. However, for Majorana neutrinos the magnetic moment bounds
are weaker than present experimental limits if $\mu_\nu$ is generated by new
physics at ~ 1 TeV, and surpass current experimental sensitivity only for new
physics scales > 10 -- 100 TeV. The discovery of a neutrino magnetic moment
near present limits would thus signify that neutrinos are Majorana particles.
