The ability to precisely tune the mechanical properties of polymeric composites is vital for harnessing these materials in a range of diverse applications. Polymer-grafted nanoparticles (PGNs) that are cross-linked into a network offer distinct opportunities for tailoring the strength and toughness of the material. Within these materials, the free ends of the grafted chains form bonds with the neighboring chains, and tailoring the nature of these bonds could provide a route to tailoring the macroscopic behavior of the composite. Using computational modeling, we simulate the behavior of three-dimensional PGN networks that encompass both high-strength "permanent" bonds and weaker, more reactive labile bonds. The labile connections are formed from slip bonds and biomimetic "catch" bonds. Unlike conventional slip bonds, the lifetime of the catch bonds can increase with an applied force, and hence, these bonds become stronger under deformation. With our 3D model, we examined the mechanical response of the composites to a tensile deformation, focusing on samples that encompass different numbers of permanent bonds, different bond energies between the labile bonds, and varying numbers of catch bonds. We found that at the higher energy of the labile bonds (U-l = 39k(B)T), the mechanical properties of the material could be tailored by varying both the number of permanent bonds and catch bonds. Notably, as much as a 2-fold increase in toughness could be achieved by increasing the number of permanent bonds or catch bonds in the sample (while the keeping other parameters fixed). In contrast, at the lower energy of the labile bonds considered here (U-l = 33k(B)T), the permanent bonds played the dominant role in regulating the mechanical behavior of the PGN network. The findings from the simulations provide valuable guidelines for optimizing the macroscopic behavior of the PGN networks and highlight the utility of introducing catch bonds to tune the mechanical properties of the system.