The phase behavior of model colloidal systems containing platelets and polymer is investigated using computer simulation and perturbation theory. For polymer coils with a diameter sP larger than 0.3sD, where sD is the diameter of the platelets, isotropic fluid-fluid coexistence is observed, in addition to the usual isotropic-nematic transition for hard disk systems. For very small polymer coils (s P<0.1sD), a nematic-nematic demixing transition is observed, although this occurs at extremely high platelet density. The case of colloidal platelets and thin stiff rods is also examined. Demixing is observed in the isotropic phase for long rods and in the nematic phase for short rods, as for large and small polymer coils, respectively. However, the nematic-nematic demixing transition for platelet-rod mixtures occurs at a much lower platelet density than in the platelet-disk mixtures.