In agreement with recently published results, when an ultra-thin (<5 nm) oxide is subjected to electrical stress, several soft-breakdown events can occur prior to the final dielectric breakdown. After the occurrence of such failure events, the current-voltage (I-V) characteristic corresponds to the superposition of highly conductive spots and background conduction through the undegraded capacitor area. In this conduction regime, the application of a low constant voltage gives rise to large leakage current fluctuations in the form of random telegraph noise. These fluctuations are demonstrated to correspond to ON/OFF switching events of one or more local conduction spots, and not to a modulation of their conductance. The experimental soft-breakdown I-V characteristics are shown to be better understood if the spot conduction is considered to be locally limited by the silicon electrodes and not by the oxide.