The impact of dark matter-neutrino interactions on the measurement of the cosmological parameters has been investigated in the past in the context of massless neutrinos exclusively. Here we revisit the role of a neutrino-dark matter coupling in light of ongoing cosmological tensions by implementing the full Boltzmann hierarchy for three massive neutrinos. Our tightest 95% CL upper limit on the strength of the interactions, parameterized via $u_chi =frac{sigma_0}{sigma_{Th}}left(frac{m_chi}{100 text{GeV}}right)^{-1}$, is $u_chileq3.34 cdot 10^{-4}$, arising from a combination of Planck TTTEEE data, Planck lensing data and SDSS BAO data. This upper bound is, as expected, slightly higher than previous results for interacting massless neutrinos, due to the correction factor associated with neutrino masses. We find that these interactions significantly relax the lower bounds on the value of $sigma_8$ that is inferred in the context of $Lambda$CDM from the Planck data, leading to agreement within 1-2$sigma$ with weak lensing estimates of $sigma_8$, as those from KiDS-1000. However, the presence of these interactions barely affects the value of the Hubble constant $H_0$.
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