Continuous flash suppression (CFS) is a potent method of inducing binocular rivalry, wherein a rapid succession of high-contrast images presented to one eye effectively blocks from awareness a low-contrast image presented to the other eye. Here we addressed whether the contents of the suppressed image can break through to awareness with extended CFS exposure. On 2/3 of the trials, we presented a faint bar (the target) to the nondominant eye while a high-contrast flickering Mondrian (the mask) was displayed to the dominant eye. Participants were first asked to report whether the target had broken through the CFS mask. Furthermore, on target-present trials, the participants were then asked to guess whether the target had appeared above or below the fixation point. In Experiment 1, the target was presented with a fixed orientation for four blocks of trials, whereas in the fifth block, the target could also have the orthogonal orientation. In Experiment 2, the target was always presented with a fixed orientation, but in the fifth block, unbeknownst to participants, the target and the mask were swapped across the eyes. We found that awareness of the target rapidly improved with training in both experiments. However, whereas Experiment 1 revealed that the improvement largely generalized across stimulus orientations, Experiment 2 showed that the effect of practice was eye-specific. The results suggest that increased breakthrough with training was due to a monocular form of learning. Finally, a control experiment was conducted to exclude the possibility that the monocular learning we reported could have been due to sensory adaptation caused by the masks.