The effect of dc electrical stress and breakdown on Josephson and quasiparticle tunneling in Nb/Al/AlOx/Nb junctions with ultrathin AlOx barriers typical for applications in superconductor digital electronics has been investigated. The junctions’ conductance at room temperature and current-voltage (I-V) characteristics at 4.2 K have been measured after the consecutive stressing of the tunnel barrier at room temperature. Electrical stress was applied using current ramps with increasing amplitude ranging from 0 to ∼1000Ic corresponding to voltages across the barrier up to ∼0.65 V, where Ic is the Josephson critical current. A very soft breakdown has been observed with polarity-dependent breakdown current (voltage). As the stressing progresses, a dramatic increase in subgap conductance of the junctions, the appearance of subharmonic current steps, and a gradual increase in both the critical and the excess currents as well as a decrease in the normal-state resistance have been observed. The observed changes in superconducting tunneling suggest a model in which a progressively increasing number of defects and associated additional conduction channels [superconducting quantum point contacts (SQPCs)] are induced by electric field in the tunnel barrier. By comparing the I-V characteristics of these conduction channels with the nonstationary theory of current transport in SQPCs based on multiple Andreev reflections by Averin and Bardas, the typical transparency D of the induced SQPCs was estimated as D∼0.7. The number of induced SQPCs was found to grow with voltage across the barrier as sinh(V/V0) with V0=0.045 V, in good agreement with the proposed model of defect formation by ion electromigration. The observed polarity dependence of the breakdown current (voltage) is also consistent with the model. Based on the observed magnitude of breakdown currents, electric breakdown of AlOx barrier during plasma processing was considered to be an unlikely cause of fabrication-induced, circuit pattern-dependent nonuniformities of Josephson junctions’ critical currents in superconductor integrated circuits.