  The introduction of Dark Matter-neutrino interactions modifies the Cosmic
Microwave Background (CMB) angular power spectrum at all scales, thus affecting
the reconstruction of the cosmological parameters. Such interactions can lead
to a slight increase of the value of $H_0$ and a slight decrease of $\sigma_8$,
which can help reduce somewhat the tension between the CMB and lensing or
Cepheids datasets. Here we show that it is impossible to solve both tensions
simultaneously. While the 2015 Planck temperature and low multipole
polarisation data combined with the Cepheids datasets prefer large values of
the Hubble rate (up to $H_0 = 72.1^{+1.5}_{-1.7} \rm{km/s/Mpc}$, when
$N_{\rm{eff}}$ is free to vary), the $\sigma_8$ parameter remains too large to
reduce the $\sigma_8$ tension. Adding high multipole Planck polarization data
does not help since this data shows a strong preference for low values of
$H_0$, thus worsening current tensions, even though they also prefer smaller
value of $\sigma_8$.
