In other words, the more "statistically significant" is the result in such an experiment, the more it is evidence for faulty measurement and against the experimenters' claim (here, FTL neutrinos).
It's still evidence for the claims, just also evidence for the experiment being faulty.
Technically, when viewed on the log-odds scale, it is exactly the same amount of evidence for either hypothesis.
On the (0;1) scale of probabilities it is, however, stronger evidence for flawed experiment. E.g. if we had a null hypothesis H0 at a prior P of 0.25 and two hypotheses H1 and H2 at 0.25 and 0.5, respectively, and we see evidence that has a likelihood of 5:1 for each of these hypotheses over H0. Then we have posterior P(H0|D)/P(H1|D)=1/5 and P(H0|D)/P(H2|D)=1/10, and after normalization, P(H0)=1/16, P(H1)=5/16, P(H2)=10/16. So, in this situation...
A mundane cause for a surprising result. Consider this unconfirmed for now, however unsurprising it sounds.
Source: Science/AAAS