Data-driven and learning-based methods have attracted considerable attention in recent years both for the analysis of dynamical systems and for control design. While there are many interesting and exciting results in this direction, our understanding of fundamental limitations of learning for control is lagging. This talk will focus on the question of when learning can be hard or impossible in the context of dynamical systems and control. In the first part of the talk, I will discuss a new observation on immersions and how it reveals some potential limitations in learning Koopman embeddings. In the second part of the talk, I will show what makes it hard to learn to stabilize linear systems from a sample-complexity perspective. While these results might seem negative, I will conclude the talk with some thoughts on how they can inspire interesting future directions.