Visually identifying materials is crucial for many tasks, yet material perception remains poorly understood. Distinguishing mirror from glass is particularly challenging as both materials derive their appearance from their surroundings, yet we rarely experience difficulties telling them apart. Here we took a 'big data' approach to uncovering the underlying visual cues and processes, leveraging recent advances in neural network models of vision. We trained thousands of convolutional neural networks on >750,000 simulated mirror and glass objects, and compared their performance with human judgments, as well as alternative classifiers based on 'hand-engineered' image features. For randomly chosen images, all classifiers and humans performed with high accuracy, and therefore correlated highly with one another. To tease the models apart, we then painstakingly assembled a diagnostic image set for which humans make highly systematic errors, allowing us to decouple accuracy from human-like performance. A large-scale, systematic search through feedforward neural architectures revealed that relatively shallow networks predicted human judgments better than any other models. However, surprisingly, no network correlated better than 0.6 with humans (below inter-human correlations). Thus, although the model sets new standards for simulating human vision in a challenging material perception task, the results cast doubt on recent claims that such architectures are generally good models of human vision.