Recent advances in Deep convolutional Neural Networks (DNNs) have enabled unprecedentedly accurate computational models of brain representations, and present an exciting opportunity to model diverse cognitive functions. State-of-the-art DNNs achieve human-level performance on object categorisation, but it is unclear how well they capture human behavior on complex cognitive tasks. Recent reports suggest that DNNs can explain significant variance in one such task, judging object similarity. Here, we extend these findings by replicating them for a rich set of object images, comparing performance across layers within two DNNs of different depths, and examining how the DNNs' performance compares to that of non-computational "conceptual" models. Human observers performed similarity judgments for a set of 92 images of real-world objects. Representations of the same images were obtained in each of the layers of two DNNs of different depths (8-layer AlexNet and 16-layer VGG-16). To create conceptual models, other human observers generated visual-feature labels (e.g., "eye") and category labels (e.g., "animal") for the same image set. Feature labels were divided into parts, colors, textures and contours, while category labels were divided into subordinate, basic, and superordinate categories. We fitted models derived from the features, categories, and from each layer of each DNN to the similarity judgments, using representational similarity analysis to evaluate model performance. In both DNNs, similarity within the last layer explains most of the explainable variance in human similarity judgments. The last layer outperforms almost all feature-based models. Late and mid-level layers outperform some but not all feature-based models. Importantly, categorical models predict similarity judgments significantly better than any DNN layer. Our results provide further evidence for commonalities between DNNs and brain representations. Models derived from visual features other than object parts perform relatively poorly, perhaps because DNNs more comprehensively capture the colors, textures and contours which matter to human object perception. However, categorical models outperform DNNs, suggesting that further work may be needed to bring high-level semantic representations in DNNs closer to those extracted by humans. Modern DNNs explain similarity judgments remarkably well considering they were not trained on this task, and are promising models for many aspects of human cognition.