In living tissues, collagen networks rarely exist alone because they are embedded within other biological matrices. When combined, collagen networks rigidify via synergistic mechanical interactions and stiffen only with higher mechanical loads. However, how cells respond to the nonlinear elasticity of collagen in hybrid networks remains largely unknown. Here, we demonstrate that when collagen rigidifies by the interpenetration of a second polymer, the amount of force that initially stiffens the network (onset of stiffening, σ) increases and is sufficient to stimulate an increase in intracellular tension. We investigated this effect by precisely controlling the nonlinear elasticity of collagen with the synthetic semiflexible polymer, polyisocyanopeptides. We find that small increases in σ induce a biphasic response in cell-matrix interactions, influencing how cells migrate, proliferate, and generate contractile force. Our results suggest that cells adaptively respond to changes in the nonlinear mechanics of collagen, which may be a mechanistic behavior used during tissue homeostasis or when collagen rigidifies during pathological conditions.