The recent high precision maps of cosmic microwave anisotropies combined with measurements of the galaxy power spectrum from new large-scale redshift surveys have allowed stringent bounds on the sum of the neutrino masses to be placed. The past analyses, however, have implicitly assumed that the spectrum of primordial density fluctuations is adiabatic and coherent, as predicted in the simplest models of inflation. In this paper, we show that the limits hold even if the assumption on the primordial power spectrum is relaxed to allow for a contribution of nonadiabatic, incoherent fluctuations such as would be predicted by topological defects.