The NA60 experiment studies muon pair production at the CERN Super Proton Synchrotron. In this Letter we report on a precision measurement of J/psi in In-In collisions. We have studied the J/psi centrality distribution, and we have compared it with the one expected if absorption in cold nuclear matter were the only active suppression mechanism. For collisions involving more than approximately 80 participant nucleons, we find that an extra suppression is present. This result is in qualitative agreement with previous Pb-Pb measurements by the NA50 experiment, but no theoretical explanation is presently able to coherently describe both results.