Under shear, a system of particles changes its contact network and becomes unstable as it transitions between mechanically stable states. For hard spheres at zero pressure, contact breaking events necessarily generate an instability, but this is not the case at finite pressure, where we identify two types of contact changes: network events that do not correspond to instabilities and rearrangement events that do. The relative fraction of such events is constant as a function of system size, pressure and interaction potential, consistent with our observation that both nonlinearities obey the same finite-size scaling. Thus, the zero-pressure limit of the nonlinear response is highly singular.