Tunable mechanics and fracture resistance are hallmarks of biological tissues whose properties arise from extracellular matrices comprised of double networks. To elucidate the origin of these desired properties, we study the shear modulus and fracture properties of a rigidly percolating double network model comprised of a primary network of stiff fibers and a secondary network of flexible fibers. We find that when the primary network density is just above its rigidity percolation threshold, the secondary network density can be tuned to facilitate stress relaxation via non-affine deformations and provide mechanical reinforcement. In contrast, when the primary network is far above its rigidity threshold, the double network is always stiff and brittle. These results highlight an important mechanism behind the tunability and resilience of biopolymer double networks: the secondary network can dramatically alter mechanical properties from compliant and ductile to stiff and brittle only when the primary network is marginally rigid. This journal is © The Royal Society of Chemistry.