The phase behavior of rod-plate mixtures was investigated using model systems of unambiguously rod- and plate-shaped colloids. Neither at equilibrium nor in the early stages of the phase separation process do we observe the theoretically disputed biaxial nematic phase. Instead, the mixtures show demixing into an isotropic and a separate rod- and a plate-dominated uniaxial nematic phase. The high-concentration regime of the rod-plate phase diagram is exceptionally rich because of the appearance of two extra liquid crystal phases: a rod-rich (presumably) nematic phase and a plate-rich columnar phase. This leads to the appearance of several four-phase equilibria and even a five-phase equilibrium. We explain the observed topology of the phase diagram by the interplay between the rod- and platelike shape of the particles and their polydispersity.