Pygmalion's answer isn't quite complete. It's not so much that recalculations were done as much as that OPERA did new calculations using new data, by comparing the timings of events in another neutrino experiment at the same site with their own, which allowed them to be fairly certain that sometime in 2008 something happened that delayed the timing signal. That didn't matter for other experiments (AFAIK) but did matter for the neutrino velocity measurements. Dilaton's link in comments to Matt Strassler's blog provides quite a lot of detail, as I'm sure Anna V's link also does.
[EDIT: As Anna points out below, one should distinguish between OPERA, which was a large experiment designed to detect neutrino oscillations, and the small sub-experiment that was added to measure the speed of light, using the same experimental apparatus for detecting neutrinos with some additional hardware thrown in.]
I presume that if someone outside OPERA hadn't tipped off the press (which is what I understand to have happened) they would not have gone public, but what is slightly scandalous is that because there was no redundancy in their timing signal path they at first had almost no idea whether their timing was correct or not. It's not enough to go around checking that all the wires are connected correctly, there has to be an audit trail, confirming that multiple signal paths give the same result. As soon as they had two ways to measure the timings, they knew there was something wrong with the timing, and that it pretty much accounted for the 60 nanoseconds. The person who figured out how to work that audit, which took three months to work through to the outside world, deserves a thank you from OPERA.
The hard question for the laboratory is whether the fact that this was meant to be an opportunistic experiment that could be done quickly and cheaply justifies the experiment going ahead without a robust design. In retrospect it's clear that with the experiment as designed, if they found neutrinos traveling at the speed of light no-one would be especially interested, but if the experiment found neutrinos traveling faster or slower they wouldn't be able to put their hands on their hearts and swear up and down that the experimental design was robust. It's cost the laboratory a lot of time, money, and prestige that they didn't think the experiment through more carefully at the design stage.
My reason for getting into this is that your question asks why we thought neutrinos travel faster than light, however no Physicist I know thought that this experiment was enough to make us think that. From a theoretical point of view, it was a startling experimental result, so that just one experiment would have to be watertight on paper to persuade theoretical physicists. A different, more careful experiment might have had more Physicists thinking more seriously that perhaps neutrinos do travel faster than light, but this wasn't that experiment.
BTW, two senior people at the laboratory have reportedly moved to different responsibilities.