Mixtures of nanoparticles (NPs) with hybridizing grafted DNA or DNA-like strands have been shown to create highly tunable NP–NP interactions, which, if designed to give nonadditive mixing, could lead to a richer self-assembly behavior. While nonadditive mixing is known to result in nontrivial phase behavior in molecular fluids, its effects on colloidal/NP materials have been much less studied. Such effects are explored here via molecular simulations for a binary system of tetrahedral patchy NPs, known to self-assemble into the diamond phase. The NPs are modeled with raised patches that interact through a coarse-grained interparticle potential representing DNA hybridization between grafted strands. It was found that these patchy NPs spontaneously nucleate into the diamond phase, and that hard-interacting NP cores eliminated the competition between the diamond and BCC phases at the conditions studied. Our results also showed that while higher nonadditivity had a small effect on phase behavior, it kinetically enhanced the formation of the diamond phase. Such a kinetic enhancement is argued to arise from changes in phase packing densities and how these modulate the interfacial free energy of the crystalline nucleus by favoring high-density motifs in the isotropic phase and larger NP vibrations in the diamond phase.