Measurements of the fluctuations in the tunneling currentItthrough a thin SiO2insulating layer, by means of an ultralow noise measurement set, showed that after a first time interval in which its power spectral density is stationary and proportional toI2t, an on‐off modulation ofItarises, just before oxide breakdown. This bistable noise seems to be related to localized phenomena controlled by trapping‐detrapping processes within the oxide. Two possible mechanisms which could give rise to this bistable noise are discussed.