Strain-stiffening behavior common to biopolymer networks is difficult to reproduce in synthetic networks. Physically associating synthetic polymer networks can be an exception to this rule and can demonstrate strain-stiffening behavior at relatively low values of strain. Here, the stiffening behavior of model elastic networks of physically associating triblock copolymers is characterized by shear rheometry. Experiments demonstrate a clear correlation between network structure and strain-stiffening behavior. Stiffening is accurately captured by a constitutive model with a single fitting parameter related to the midblock length. The same model is also effective for describing the stiffening of actin, collagen, and other biopolymer networks. Our synthetic polymer networks could be useful model systems for biological materials due to (1) the observed similarity in strain-stiffening behavior, which can be quantified and related to network structure, and (2) the tunable structure of the physically associating network, which can be manipulated to yield a desired response.