Many surface cues support three-dimensional shape perception, but humans can sometimes still see shape when these features are missing—such as when an object is covered with a draped cloth. Here we propose a framework for three-dimensional shape perception that explains perception in both typical and atypical cases as analysis-by-synthesis, or inference in a generative model of image formation. The model integrates intuitive physics to explain how shape can be inferred from the deformations it causes to other objects, as in cloth draping. Behavioural and computational studies comparing this account with several alternatives show that it best matches human observers (total n = 174) in both accuracy and response times, and is the only model that correlates significantly with human performance on difficult discriminations. We suggest that bottom-up deep neural network models are not fully adequate accounts of human shape perception, and point to how machine vision systems might achieve more human-like robustness.