  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, generated by physics above the electroweak scale. For
Majorana neutrinos, these bounds are weaker than present experimental limits if
$\mu_\nu$ if generated by new physics at $\sim$ 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.
