We investigate theoretically the origins of observed variations in the critical currents of Nb/Al-AlOx/Nb Josephson junctions in terms of various types of disorder. We consider the following disorder sources: vacancies within the Al layer; thickness variations in the AlOx layer; and "pinholes" (i.e., point contacts) within the AlOx layer. The calculations are all performed by solving the microscopic Bogoliubov-de Gennes Hamiltonian self-consistently. It is found that a small concentration of vacancies within the Al layer is sufficient to suppress the critical current, while the presence of a small number of thick regions of the oxide layer induces a similar effect as well. The pinhole scenario is found to result in anomalous behavior that resembles neither that of a pure tunnel junction nor that of an SNS junction, but a regime that interpolates between these two limits. We comment on the degree to which each of the three scenarios describes the actual situation present in these junctions. arXiv:1908.08360v1 [cond-mat.supr-con]

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