Many metals display resistivity saturation - a substantial decrease in the slope of the resistivity as a function of temperature, that occurs when the electron scattering rate becomes comparable to the Fermi energy; this is know as the Mott-Ioffe-Regel (MIR) limit. We present a tractable model of a large N number of electronic bands coupled to N² optical phonon modes, which displays resistivity saturation near the MIR limit, a value that emerges naturally. The saturation is due to the dominance, at high temperatures, of a saturating conduction channel; this behavior is reminiscent of the phenomenological parallel resistor formula. The high temperature non-Boltzmann regime is interpreted by through the Einstein relation.