Are (feedforward) convolutional neural networks (CNNs) good models for the human visual system? Here, we used visual crowding as a well-controlled psychophysical test to probe CNNs. Visual crowding is a ubiquitous breakdown of object recognition in the human visual system, whereby targets become jumbled and unrecognisable in the presence of flanking objects. Humans exhibit several well-documented effects of crowding, such as invariance to size, where the size of the target and flanker letters may be changed without impacting the strength of crowding. We show that feedforward CNNs are unable to reproduce invariance to size, confusion between target and flanker identities, and importantly uncrowding, where paradoxically increasing the number of flankers improves performance. We investigate this phenomenon using a recurrent, neurally inspired model called LAMINART, which we find can reproduce uncrowding as observed in humans. Furthermore, we show that capsule networks, a recurrent family of CNNs with grouping and segmentation mechanisms, outperform any other models of uncrowding to date, demonstrating the importance of grouping and segmentation in mechanisms in visual information processing in general.