  We analyze the implications of neutrino masses for the magnitude of neutrino
magnetic moments. By considering electroweak radiative corrections to the
neutrino mass, we derive model-independent naturalness upper bounds on neutrino
magnetic moments, $\mu_\nu$, 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 contribution to the mass is Yukawa suppressed. The bounds we derive for
magnetic moments of Majorana neutrinos 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.
