We use computer simulations to study the existence and stability of a biaxial nematic N_{b} phase in systems of hard polyhedral cuboids, triangular prisms, and rhombic platelets, characterized by a long (L), medium (M), and short (S) particle axis. For all three shape families, we find stable N_{b} states provided the shape is not only close to the so-called dual shape with M=sqrt[LS] but also sufficiently anisotropic with L/S>9,11,14,23 for rhombi, (two types of) triangular prisms, and cuboids, respectively, corresponding to anisotropies not considered before. Surprisingly, a direct isotropic-N_{b} transition does not occur in these systems due to a destabilization of N_{b} by a smectic (for cuboids and prisms) or a columnar (for platelets) phase at small L/S or by an intervening uniaxial nematic phase at large L/S. Our results are confirmed by a density functional theory provided the third virial coefficient is included and a continuous rather than a discrete (Zwanzig) set of particle orientations is taken into account.