Frictional contact has been extensively studied as the core underlying behavior of legged locomotion and manipulation, and its nearly-discontinuous nature makes planning and control difficult even when an accurate model of the robot is available. Here, we present empirical evidence that learning an accurate model in the first place can be confounded by contact, as modern deep learning approaches are not designed to capture this non-smoothness. We isolate the effects of contact's non-smoothness by varying the mechanical stiffness of a compliant contact simulator. Even for a simple system, we find that stiffness alone dramatically degrades training processes, generalization, and data-efficiency. Our results raise serious questions about simulated testing environments which do not accurately reflect the stiffness of rigid robotic hardware. Significant additional investigation will be necessary to fully understand and mitigate these effects, and we suggest several avenues for future study.