The measured speed of neutrinos can't be separated from $c$ within experimental limits. The standard model includes no mechanism to grant neutrinos mass. The mass of neutrinos can't be detected. Nevertheless, if you were to conclude that neutrinos are massless you would be wrong (as they experience subjective time and so can't travel at $c$).
Can we falsify the hypothesis that photons have a tiny mass and travel just a little bit slower than $c$? What's the largest mass photons could have?