In a nutshell:
Gravitational fields cause time-dilation. Therefore clocks in a gravity field "run slow". The other type of effect to look out for is due to velocity...
So the clocks were synchronized and then the 1 was transported from point A to point B. Point A and B being the start and end of the journey the neutrinos would undertake.
The speed of the transporting vehicle was ' << c' , so no relativistic effects were involved there. However, the clock was transported over drastic altitude varations. This implies the gravitational field wasn't constant and would give rise to the clock (as it ascends) to run faster than its 'synchronized' counterpart.
You may say that the G-field of the earth is too small to account for this, BUT
Relativistic effects due to varying G-field can be measured for EVEN A DIFFERENCE OF 1CM(!).
=> clocks not properly synchronized
=> measurement error for the speed of the neutrino
=> people acting all crazy even though they didn't understand Relativity in the first place
EDIT: half my original post disappeared(?) hmf.
