Using a straightforward extension of the Onsager-theory for hard rods, we consider the thermodynamic stability of the isotropic (I) and nematic (N) phase of binary mixtures of thick and thin hard rods of the same length. We show that such mixtures not only exhibit the expected I-N ordering transition and the previously predicted depletion driven I-I demixing transition, but also a N-N demixing transition driven by the orientation entropy of the thinner rods. For various values of the diameter ratio of the two species we present the phase diagrams, which exhibit I-N, I-I and N-N coexistence, I-N-N and I-I-N triple points, and I-I and N-N critical points. We also present the results of computer simulations of the I-I and I-N coexistence for diameter ratio 1:10, which are in good agreement with the theory.