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, $mu_ u$, 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 $mu_ u$ 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.
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