Most computational theories of stereopsis require a registration stage prior to stereo matching to reduce the matching to a one-dimensional search. Even after registration, it is critical that the stereo matching process tolerate some degree of residual misalignment. In this paper, we study with psychophysical techniques the tolerance to vertical disparity in situations in which false targets abound ?? in random dot stereograms ??d eye movements are eliminated. Our results show that small amounts of vertical disparity significantly impair depth discrimination in a forced-choice task. Our main results are: a) vertical disparity of only the central "figure" part of a random dot stereogram can be tolerated up to about 3.5', b) vertical disparity of the "figure + ground" is tolerated up to about 6.5', and c) the performance of the Grimson implementation of the Marr-Poggio stereo matching algorithm for the stereograms of experiment (a) is consistent with the psychophysical results. The algorithm's tolerance to vertical disparity is due exclusively to the spatial averaging of the underlying filters. The algorithm cannot account by itself for the results of experiment (b). Eye movements, which are the principal registration mechanism for human stereopsis, are accurate to within about 7'. Our data suggest that tolerance to this residual vertical disparity is attained by two non-motor mechanisms: 1) the spatial average performed by the receptive fields that filter the two images prior to stereo matching, and 2) a non-motor shift mechanism that may be driven at least in part by monocular cues.