Variations in the predominance of an object engaged in binocular rivalry may arise from variations in the durations of dominance phases, suppression phases, or both. Earlier work has shown that the predominance of a binocular rival target is enhanced if that target fits well-via common color, orientation, or motion-with its surrounding objects. In the present experiments, the global context outside of the region of rivalry was changed during rivalry, to learn whether contextual information alters the ability to detect changes in a suppressed target itself. Results indicate that context will maintain the dominance of a rival target, but will not encourage a suppressed target to escape from suppression. Evidently, the fate of the suppressed stimulus is determined by neural events distinct from those responsible for global organization during dominance. To reconcile diverse findings concerning rivalry, it may be important to distinguish between processes responsible for selection of one eye's input for dominance from processes responsible for the implementation and maintenance of suppression.