Well, not whilst this is only operating at a neutrino level, I would have thought. Whether such a trick could be applied to larger objects is both unknown and, on first glance, probably unlikely. What we are seeing here, most likely, is a quantum level effect, and quantum, like biology, doesn't scale up to macro sizes very well. Make an exact duplicate of a pigeon, only ten times bigger, and it will break its legs every time it lands. I think you'll come across a similar type of problem with this.
Edit: At the end of the day, it depends how fundamental the flaw is in the original theories I suppose (assuming this neutrino event stands up to peer-review), but I still consider relativity to be a pretty sound theory, it's probably a case of a slight alteration than a complete re-write, and it all depends on what alteration it involves as to what impact it has.
Edit 2 : I suppose the next question Physicists will be asking themselves is 'does the neutrino actually occupy real-space for the entire journey, or is there some kind distant relative to electron state-jumping going on here?' Since moving the neutrino out of the 'Newtonian universe' would allow it to do this without actually breaking any relativity laws whatsoever.
