We cannot guarantee that training datasets are representative of the distribution of inputs that will be encountered during deployment. So we must have confidence that our models do not over-rely on this assumption. To this end, we introduce a new method that identifies context-sensitive feature perturbations (e.g. shape, location, texture, colour) to the inputs of image classifiers. We produce these changes by performing small adjustments to the activation values of different layers of a trained generative neural network. Perturbing at layers earlier in the generator causes changes to coarser-grained features; perturbations further on cause finer-grained changes. Unsurprisingly, we find that state-of-the-art classifiers are not robust to any such changes. More surprisingly, when it comes to coarse-grained feature changes, we find that adversarial training against pixel-space perturbations is not just unhelpful: it is counterproductive.