We study the phase behavior of hard rods with length polydispersity, within a simplified Onsager model. We focus on unusual effects occurring when the rod length distribution has a "fat" (e.g., log-normal) tail, up to some cutoff. The relatively large number of long rods in the system has strong effects: the isotropic cloud curve, which defines where a nematic phase first occurs as density is increased, exhibits a kink; the properties of the coexisting nematic shadow phase change discontinuously there. A narrow three-phase isotropic-nematic-nematic coexistence region exists near the kink in the cloud curve, even though the length distribution is unimodal. A theoretical derivation of the isotropic cloud curve and nematic shadow curve is sketched for the limit of large cutoff. The two curves are shown to collapse onto each other in the limit. The coexisting isotropic and nematic phases are essentially identical, the only difference being that the nematic contains a larger number of the longest rods; also, only long rods show significant nematic ordering. (C) 2003 American Institute of Physics.