Model physically associating solutions of acrylic triblock copolymer molecules in a midblock-selective solvent displayed nonlinear strain-stiffening behavior which transitioned to rapid strain softening during shear start-up experiments at reduced rates spanning almost 4 orders of magnitude. Softening was believed to result from the shear-induced formation of highly localized regions of deformation in the macromolecular network. This behavior was accurately captured by a model that incorporated the strain energy and relaxation behavior of individual network strands in the solution. Flow curves predicted from the model were nonmonotonic, consistent with the onset of flow instabilities at high shear rates. The nonlinear stress response reported here, coupled with the wide range of accessible relaxation times of these thermoreversible solutions, makes them ideal model systems for studies of failure-mode transitions in physically associating solutions and gels.