We present a microscopic model for systems showing resistivity saturation. An essentially exact quantum Monte Carlo calculation demonstrates that the model describes saturation. We give a simple explanation for saturation, using charge conservation and considering the limit where thermally excited phonons have destroyed the periodicity. Crucial model features are phonons coupling to the hopping matrix elements and a unit cell with several atoms. We demonstrate the difference to a model of alkali-doped C60 with coupling to the level positions, for which there is no saturation.