Humans learn categories of complex objects quickly and from a few examples. Random projection has been suggested as a means to learn and categorize efficiently. We investigate how random projection affects categorization by humans and by very simple neural networks on the same stimuli and categorization tasks, and how this relates to the robustness of categories. We find that (1) drastic reduction in stimulus complexity via random projection does not degrade performance in categorization tasks by either humans or simple neural networks, (2) human accuracy and neural network accuracy are remarkably correlated, even at the level of individual stimuli, and (3) the performance of both is strongly indicated by a natural notion of category robustness.