Important insights towards the explainability of neural networks reside in the characteristics of their decision boundaries. In this work, we borrow tools from the field of adversarial robustness, and propose a new perspective that relates dataset features to the distance of samples to the decision boundary. This enables us to carefully tweak the position of the training samples and measure the induced changes on the boundaries of CNNs trained on large-scale vision datasets. We use this framework to reveal some intriguing properties of CNNs. Specifically, we rigorously confirm that neural networks exhibit a high invariance to non-discriminative features, and show that the decision boundaries of a DNN can only exist as long as the classifier is trained with some features that hold them together. Finally, we show that the construction of the decision boundary is extremely sensitive to small perturbations of the training samples, and that changes in certain directions can lead to sudden invariances in the orthogonal ones. This is precisely the mechanism that adversarial training uses to achieve robustness.