I was brought up short by a news thing about neutrinos apparently measured faster than light. The comments in that link are sad sad sad. But to be fair, it's a poorly understood part of science.
My favorite description of relativity vis a vis FTL, with the problems this poses. Basically, special relativity manages to be internally consistent as long as nothing goes faster than light (in a vacuum) or if it does, it is restricted in how it does so. Of course, special relativity could be wrong.
In quantum mechanics, the things we are pleased to think of as the Laws of Physics are more like "firm guidelines" which are only obeyed on average. This means that the speed of light is approximate and energy violations occur all the time. There are still restrictions on how this can happen though, so that the physics does not contradict itself.
An example is the FTL information transfer apparent in Quantum Entanglement experiments (used to be called "teleportation"). In these experiments, an observation made in one place affects the results of an observation in another place in such a way that it cannot be turned into a kind of FTL radio.
An easier example would be the spot of light cast from (say) a laser pointer onto a surface. If you move the pointer through an angle A, the spot moves RA where R is the distance to the surface. Make R big enough and even quite slow movements of the pointer results in FTL movement of the spot.
But these are neutrinos... weakly interacting particles with a very small mass (<16eV/c^2) so we need to be more careful about what we actually say is happening.
The researchers set up an experiment to test the quantum oscillation of mu and tau neutrinos (mu and tau neutrinos can be set up to change between each other cyclically) from a well configured source. This sort of thing involves setting up a source of mu neutrinos with a known set of statistics, firing them off at a target, and counting how many mu and tau neutrinos get detected (compensating for the sensitivity of the detector and the background of detection events) to see how the result compares with the theoretically projected counts.
Presumably they got fewer tau and more mu than expected, which could be interpreted as a higher than expected speed (or a slower than expected oscillation). If you have reason to discount the oscillation rate as a cause, then you can use the data to compute what sort of speed must have been achieved ... they did this and reported that the anomalous neutrinos arrived at the detector 60ns sooner than light would have traversed the same distance in a vacuum. The uncertainty in the calculation was 10ns - so the result is highly significant, statistically.
At this stage, the findings have been made public so the scientific community can check them, maybe repeat the experiments. This was the right thing to do. It shows scientific humility to put your findings up to scrutiny like this ... someone could find an embarrassing mistake.
I stress: I don't know for sure that this is the method they used. I am going by how such experiments are _usually_ done. Maybe someone was sitting on the detector with a stop-watch, but it's unlikely.
One of the big problems is that there have been plenty of opportunities to see something like this before now. For example, the neutrinos from SP1987A arrived about 3 hours ahead of the light-pulse from over 187000ly away. This difference is consistent with the light having to traverse the surrounding stellar material first. If the above figures are treated as correct, then they should have been of the order of months early, even taking into account the uncertainty in the distance.
There have been many neutrino experiments, and many experiments looking specifically for this sort of thing using other particles.
One of the ways that has been considered for FTL concerns the way QM represents particles as waves. In _wave mechanics_, a particle is represented by a wave-packet - the square of the part of the wave-packet inside the detector is the probability of the particle being detected. In wave-mechanics, it is these wave-packets that move around.
A typical wave packet used in an experiment is Gaussian (normal distribution). Relativity means that the speed of the peak of the packet (the group velocity) cannot be faster than light. But, with this kind of packet, different parts of it travel at different speeds. In particular, the leading parts are faster and the trailing parts are slower. So it is technically possible to set up a situation where some distant part of the leading edge is moving faster than light. Thus, there is a small, but non-zero, chance that the particle gets detected well ahead of when the peak would have arrived at the detector. This would detect as FTL.
To my knowledge, this has not been seen.
It could be something like this, with the particular care taken in these experiments being the reason nobody saw it before now. However, this is highly unlikely. A fluctuation in the background mu neutrinos would also produce this sort of result, much like you can get a freak wave when you are out rock-fishing. This is why it is very important to repeat these experiments.