Quote - Haven't atomic clocks been used to verify relativistic effects in different rates of the passages of time at different altitudes on Earth, as well?  Could there be some other explanation for this kind of thing, if one type of particle exceeding the speed limit fells relativity altogether?  Or would that be true?  Would one deviant particle type invalidate relativity?  (And whatever happened with the idea that tachyons move backwards in time?  Invalid, still open?  Would that break the current theory?)

Atomic clocks have been flown on satellites and aircraft to demonstrate relativistic time dilation. Another experimental proof that people might not have heard of involves cosmic rays. When they hit the atmosphere, they produce the same sort of effect humans create in particle colliders, that is, a shower of particles resulting from the collision. Some of the these particles are extremely short-lived, so short that they shouldn't be detectable from the Earth's surface, but they are...because their velocity,  causes time dilation, slowing their decay rate enough that detectors on the ground can see them before they decay away.

The same thing is seen in particle colliders. The Large Hadron Collider accelerates particles to slightly less than the speed of light in a vacuum, causing a dilation factor of around 7500: that is, a particle which at relatively slow speeds should decay in 1 nanosecond would take about 7500 nanoseconds to decay at that velocity. Again, experimentally confirmed.

As for the neutrinos being discussed, the current betting right now is that there's some kind of experimental error that will be shown to have occurred.