Feedforward Convolutional Neural Networks (ffCNNs) have become state-of-the-art models both in computer vision and neuroscience. However, human-like performance of ffCNNs does not necessarily imply human-like computations. Previous studies have suggested that current ffCNNs do not make use of global shape information. However, it is currently unclear whether this reflects fundamental differences between ffCNN and human processing or is merely an artefact of how ffCNNs are trained. Here, we use visual crowding as a well-controlled, specific probe to test global shape computations. Our results provide evidence that ffCNNs cannot produce human-like global shape computations for principled architectural reasons. We lay out approaches that may address shortcomings of ffCNNs to provide better models of the human visual system.