We study the combined effect of nuclear absorption and final state interaction with co-moving hadrons on the J/psi and psi' suppression in proton-nucleus and nucleus-nucleus collisions. We obtain a reasonable description of the Pb-Pb data using theoretically meaningful values of the cross-sections involved and without introducing any discontinuity in the J/psi or psi' survival probabilities. However, the description of both pA and SU data is significantly worse than the one obtained with nuclear absorption alone.