The NA60 experiment at the CERN SPS has measured low mass dimuons from p-A and In-In collisions with a mass resolution of about 2% for the light vector mesons and good acceptance down to low pair pt. This paper focuses on a net data sample of 330000 pairs from In-In, about one half of the total collected. The data for p-A and peripheral In-In compare very well to the cocktail of neutral meson decays, with production yields in agreement with previous measurements. In more central In-In collisions, however, a significant excess is seen over the expected reference, which increases with event multiplicity and is highest for low pt pairs. We compare our data to the superposition of meson decays and theoretical results on thermal radiation from pion annihilation in the hot fireball, including or not medium modifications of the intermediate rho meson. Of the two standard medium effects, broadening or mass shift of the rho, one is clearly favored.