We consider the phenomenon of adversarial examples in ReLU networks with independent Gaussian parameters. For networks of constant depth and with a large range of widths (for instance, it suffices if the width of each layer is polynomial in that of any other layer), small perturbations of input vectors lead to large changes of outputs. This generalizes results of Daniely and Schacham (2020) for networks of rapidly decreasing width and of Bubeck et al (2021) for two-layer networks. Our proof shows that adversarial examples arise in these networks because the functions they compute are \emph{locally} very similar to random linear functions. Bottleneck layers play a key role: the minimal width up to some point in the network determines scales and sensitivities of mappings computed up to that point. The main result is for networks with constant depth, but we also show that some constraint on depth is necessary for a result of this kind, because there are suitably deep networks that, with constant probability, compute a function that is close to constant.