We have studied a system of polydisperse, charged colloidal gibbsite platelets with a bimodal distribution in the particle aspect ratio. We observe a density inversion of the coexisting isotropic and nematic phases as well as a three-phase equilibrium involving a lower density nematic phase, an isotropic phase of intermediate density, and a higher density columnar phase. To relate these phenomena to the bimodality of the shape distribution, we have calculated the liquid crystal phase behavior of binary mixtures of thick and thin hard platelets for various thickness ratios. The predictions are based on the Onsager-Parsons theory for the isotropic-nematic (I-N) transition combined with a modified Lennard-Jones-Devonshire cell theory for the columnar (C) state. For sufficiently large thickness ratios, the phase diagram features an I-N density inversion and triphasic I-N-C equilibrium, in agreement with experiment. The density inversion can be attributed to a marked shape fractionation among the coexisting phases with the thick species accumulating in the isotropic phase. At high concentrations, the theory predicts a coexistence between two columnar phases with distinctly different concentrations. In experiment, however, the demixing transition is pre-empted by a transition to a kinetically arrested, glassy state with structural features resembling a columnar phase.