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 $|mu_ u| < 10^{-14}$ Bohr magnetons. This bound is several orders of magnitude stronger than those obtained from solar and reactor neutrino data and astrophysical observations.
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