We present self-consistent kinetic simulations of the electron response to finite duration shear Alfvén wave pulses in a magnetized plasma. In Earth's magnetosphere, the evidence suggests that parallel electric fields in inertial scale shear Alfvén waves can accelerate electrons in the geomagnetic field-aligned direction. Here, we study large-amplitude wave forms as they travel through ambient plasma at phase velocities which are consistent with resonant electron acceleration predicted by linear kinetic theory. Our self-consistent simulations reveal that the wave potential evolves nonlinearly as shear Alfvén wave pulses travel through the simulation domain. The evolution of the wave pulse from a symmetrical to a nonsymmetrical potential structure, and the large perturbation in the distribution function required to carry the parallel current of the pulse, leads to nonresonant acceleration of electrons (i.e., acceleration of electrons which are not traveling at the same velocity as the wave). We compare the signature of resonant and nonresonant electron acceleration with data from a low-altitude spacecraft and suggest an explanation for features often referred to as field-aligned suprathermal electron bursts. Finally, we discuss how resonant and nonresonant acceleration of electrons is affected by the perpendicular wavelength and amplitude of shear Alfvén wave pulses.