Recently, deep networks have achieved impressive semantic segmentation performance, in particular thanks to their use of larger contextual information. In this paper, we show that the resulting networks are sensitive not only to global adversarial attacks, where perturbations affect the entire input image, but also to indirect local attacks, where the perturbations are confined to a small image region that does not overlap with the area that the attacker aims to fool. To this end, we introduce an indirect attack strategy, namely adaptive local attacks, aiming to find the best image location to perturb, while preserving the labels at this location and producing a realistic-looking segmentation map. Furthermore, we propose attack detection techniques both at the global image level and to obtain a pixel-wise localization of the fooled regions. Our results are unsettling: Because they exploit a larger context, more accurate semantic segmentation networks are more sensitive to indirect local attacks. We believe that our comprehensive analysis will motivate the community to design architectures with contextual dependencies that do not trade off robustness for accuracy.