There are an awful lot of very smart people who have participated (all 174 of them, as listed on the research paper), but clearly they don't believe it either, which is why they are asking for more eyes. Yes, Einstein could be wrong. His work is based on application of very clever thought experiments, and although it's matched extremely well to reality thus far, it isn't impossible to be overturned. Although he has been proven right time and time again in every experimental test to date, uncovering discrepancies in existing theories is what drives forward scientific knowledge. Maybe this result will prove or disprove one of the various flavors of string theory!
On the other hand, it's probably quite likely that it will again be upheld. Just for fun, I read through the entire published paper that is the actual starting point for all the uproar. I'm only an amateur scientist, not a real one, but I did identify a couple areas that are definite assumptions to be double-checked.
(This diagram is from the original paper.)
In a nutshell: they make neutrinos at CERN in Geneva, Switzerland aimed at a detection facility called OPERA in Gran Sasso, Italy. They compute the neutrino velocity by computing the travel time divided by the distance, and to make the claims they do, they must be exceedingly accurate in measuring both. It's not like you can drop a flag, let a neutrino and photon race, and then hit a time clock. The event happens so fast and the sites are 730 km apart, the way they measure the neutrino speed is by computing an absolute time on neutrino strength waveforms from the generator and detector, and doing computation after-the-fact to synchronize the waveforms and subtract the timestamps. They then divide the time difference by the distance difference to arrive at the speed.
The devil is in the details, as they say. Here are several thoughts that occurred to me while reading the paper on what could explain how this faster-than-light measurement isn't actually real.
1) The generation and measurement systems uses hundreds of components, requiring measurement of each length of cable and computing in all the factors that add to delay--the methodology is really quite impressive. However, with all the pieces involved, there are clearly places that errors could be introduced. They've worked for years refining the measurements of each portion. Having spent my embedded career measuring milliseconds and microseconds, it's easy for me to imagine an unanticipated error of 60 nanoseconds creeping into the measurement. One single missed or miscalculated factor could account for the difference.
2) The absolute timestamps in both location must be completely in sync. They are using GPS satellites to compute time. Having done GPS timestamp work for time servers before, I have a hard time believing each is 100% identical, even after all the error correction and refinement is added. Even if, there still is the matter of Einstein's special theory of relativity itself that compounds things. Time is not absolute--it changes depending on both the pull of gravity on, and the speed of, the observer. I don't know if the two time locations account for their relative circular motion of the planet spinning, or if the gravitational field differs, for example. Maybe the difference is inconsequential, but all you need to account for is 60 billionths of a second.
3) The distance between the two sites must be precisely measured, and it is computed as line-of-sight travel through the earth. There's no way to send a photon on the same trip--it's through solid rock. So you have to know the straight line distance exactly. But the earth is not a sphere--it's slightly pear shaped, and varies from place to place. As far as I can figure, it takes takes a distance of 17 meters to account for 60 nanoseconds difference in the measurement. Knowing a latitude, longitude and altitude exactly does not let you precisely position your location. The study does use the latest very precise geodesic measurements, but an error of 17 meters between 720 kilometers seems within the realm of possibility.
4) Both the locations and the times are dependent on GPS satellite measurements. GPS was originally designed for military use. Remember years ago when the US government reduced the fudge factor on measurements to allow civilian use to improve accuracy? I think there could there be an unaccounted dependency between the two.
I'm sure that this research has been done to a level of perfection and detail that I can't imagine. Who am I to argue with 174 PhD's? But when it comes to laying bets, I'm going to put my money on Einstein and the decades of experimental validation he's got behind him.