Despite clear computational advantages in building robust neural networks, adversarial training (AT) using single-step methods is unstable as it suffers from catastrophic overfitting (CO): Networks gain non-trivial robustness during the first stages of adversarial training, but suddenly reach a breaking point where they quickly lose all robustness in just a few iterations. Although some works have succeeded at preventing CO, the different mechanisms that lead to this remarkable failure mode are still poorly understood. In this work, however, we find that the interplay between the structure of the data and the dynamics of AT plays a fundamental role in CO. Specifically, through active interventions on typical datasets of natural images, we establish a causal link between the structure of the data and the onset of CO in single-step AT methods. This new perspective provides important insights into the mechanisms that lead to CO and paves the way towards a better understanding of the general dynamics of robust model construction.