A model of the diffractive production of multipion states on hydrogen and heavy nuclei is presented. The model is statistical in nature and incorporates the effects due to the competition between many open channels at high mass. In this picture, multipion states are produced in a two-stage process: An excited meson state (fireball) is produced peripherally, through Pomeranchukon exchange, which then proceeds to decay in a manner described by the strong statistical bootstrap. As an application of this model, we study the production of three and five charged pions in pion-induced reactions on hydrogen, deuterium, and heavier nuclei. In particular, the size of the cross section for diffractively producing five charged pions is related uniquely, through factorization, to that for producing three charged pions. The results, including mass distributions, are in good agreement with the available data.