In this Letter, we show that upper limits on the neutrino mass translate into upper limits on the class of neutrino-matter interactions that can generate loop corrections to the neutrino mass matrix. We apply our results to µ and pi decays and derive model-independent limits on six of the ten parameters used to parametrize contributions to µ decay that do not belong to the standard model. These upper limits provide improved constraints on the five Michel parameters, rho,xi[prime],xi[prime][prime],alpha,alpha[prime], that exceed Particle Data Group constraints by at least one order of magnitude. For pi0-->nu[overline nu] we find, for the branching ratio, B(pi0-->nu[overline nu])<10^-10
