I'm putting my money on instrumentation error, or some other unforeseen flaw that skewed the measurements.
Almost certainly.
That said, the Neutrino is a bastard of a particle to detect, let alone to measure. The facilities used to detect single photon events (caused by Neutrino interaction) is massive (here is another one in Japan). The signal to noise ratio would be pretty poor, so the amount of rubbish one needs to filter through in the data is not a walk in the park.
While they use photon counting detectors, neutrino detection is not a single photon event, and the SNR is actually quite high. Almost all of the neutrinos pass through the detector completely unnoticed. However, when one hits an atomic nucleus, it generates a high energy muon. The muon is traveling very close to c and makes a whole bunch of Cherenkov radiation. This creates a line source of photons which are detected by photomultipliers. For a single neutron, many photons are detected, so the the dark counts of the PMTs are negligibile. By mapping the track back to its origin you can compute the point of impact and energy with high accuracy. The sources of noise are basically neutrinos from other sources and nuclear decay from unstable isotopes. If the source is not far enough underground, cosmic rays are another source of noise. Generally these event have different energy than the neutrinos you are looking for, so they can be excluded easily.
It's amusing to see people so attached to the idea that C should be an absolute constant.
It was only a theory in the first place.
"It is only a theory" is a total bullshit statement that has no meaning except to make the person saying it feel smarter. The significance of c is backed up by decades of experimental evidence. Throwing that out at the first reported observation would be foolishness of the highest order. This is definitely interesting work, and these guys are not just cranks like you sometimes see -- they are legitimate scientists who have data they can't explain. That doesn't mean nobody can explain it, or that there is no explanation.
For a counterpoint, consider this. According to these measurements, the neutrinos in question travel at 1.000025 c. If that were true in general, the neutrinos from a distant supernova would arrive much earlier than the light -- on the order of 4 years earlier for supernova 1987a. In reality, the neutrinos arrive a few hours early. This is explainable by the fact that the neutrinos travel from the core of the supernova almost unhindered, while the shockwave takes some time to reach the surface and create the visible response. For more distant supernovas, the time lag would be even greater, but this is not observed. This is fairly strong evidence that neutrinos do not travel measureably faster than c. For both of these measurements to be correct, either there is some unexplained dependence on the neutrino energy, neutrinos travel faster through rocks than vacuum, or there is some unknown compensating effect that slows supernova neutrinos down.