One interesting
suggestion by Ethan Siegel is that the result could be wrong simply because they're measuring a biased sample of neutrinos.
To expand a bit, it's not possible to detect each individual neutrino that they're firing through their big tunnel. Neutrinos only interact very weakly with normal matter, so the overwhelming majority of them pass harmlessly on their way, without making any noticeable impact on the detector in Italy.
In fact, the number of neutrinos detected represents approximately 0.00000000000001% of the total created. If the earliest arriving neutrinos are preferentially detected, for some reason, then your sample is biased and they appear to be travelling faster than they, in fact, are.