[AndreiVajnaII]

The theory of relativity is based on the fact that the speed of light is the same for any observer, no matter if that observer is standing still or moving. From this supposition, the rest of the theory is just mathematics. So if there is something than can move faster than that speed, then the theory will still hold, as long as everybody observes the same speed, in any state of movement they are.

[hexagonc]

The problem is that we already have previous measurements of the speed of light (in a vacuum) to high precision and they have all been roughly the same. Now I haven't read the paper but the surprise of this latest experiment is that they are measuring the speed of a particle as faster than previous measurements of the speed of light. If the latest measurements are accurate then that implies that it is possible for particles to move faster than the speed of light, or the speed of light has changed, or previous measurements were worse than expected. The first or the second possibilities would both violate special relativity.

Also, if I recall my college physics correctly, special relativity not only predicts that the speed of light will be the same for all observers but that it will also have a particular value, based on some physical constants related to electromagnetism.

[AndreiVajnaII]

No, like I said, the special theory of relativity is based on the speed of light being the same for all observers, not on its exact value. That is a supposition on which the theory is based, not one of its predictions.

If, for example we now find out that neutrinos move with 1.0002% of the speed of light, but that their speed is the same, measured by any observer, we will just replace the speed of light constant in the special relativity formulas with the speed of neutrinos constant.

But that all depends what the interpretations of the results are going to be.