Local disparity signals must interact in visual cortex to represent boundaries and surfaces of three-dimensional (3D) objects. We investigated how disparity signals interact in 3D contours and in 3D surfaces generated from the contours. We compared flat (single disparity) stimuli with curved (multi-disparity) stimuli. We found no consistent differences in sensitivity to contours vs. surfaces; for equivalent amounts of disparity, however, observers were more sensitive to flat stimuli than curved stimuli. Poor depth sensitivity for curved stimuli cannot be explained by the larger range of disparities present in the curved surface, nor by disparity averaging, nor by poor sensitivity to the largest disparity in the stimulus. Surprisingly, sensitivity to surfaces curved in depth was improved by removing portions of the surface and thus removing disparity information. Stimulus configuration had a profound effect on stereo thresholds that cannot be accounted for by disparity-energy models of V1 processing. We suggest that higher-level 3D contour or 3D shape mechanisms are involved.