Einstein is rolling over in His Grave, or Cern makes a big mistake

Then our understanding about a thousand other subjects suddenly needs to be rewritten. The reason we know c so accurately isn;t because we have super-good clocks and speedometers, it's because we can accurately measure other phenomenon than just racing photons, phenomenon like the energy released in matter-antimatter reactions. If c were wrong, those results (and thousands of others) would have shown us by now.

Neutrino mass is less important than their lack of a charge. Most matter interactions, after all, are electromagnetic repulsion between electrons in atomic orbits.Since atoms are mostly just empty space and neutrinos have no charge to attract or repel them from either the electron shell or the protons in the nucleus, they just pass right on through.The low mass of a neutrino does mean that it can move faster. Accelerating a non-zero mass toward the speed of light requires more and more energy scaling to infinity, so you can never quite get there, but the low mass of a neutrino lets it get pretty close.Photons, of course, are absorbed by atoms and re-emitted (in transparent substances anyway), and this slows light down (even though the actual photons always move at c, every time they strike another atom there's a delay as it's absorbed and re-emitted). It's why c is specifically the speed of light in a vacuum.