The theory-to-practice gap describes the discrepancy between the parametric complexity and the sampling complexity of learning. For a set of functions with a fixed parametric complexity, or the best possible approximation error rate with respect to the number of model parameters, we examine the corresponding sampling complexity, or the best error rate with respect to the number of data samples given a reconstruction algorithm. In some cases, such as polynomial or trigonometric polynomial approximation, these two rates coincide: determining N degrees of freedom requires N point samples. However, for approximation spaces of neural networks, this intuition does not hold; indeed, the sampling complexity rate is fundamentally limited. As the parametric complexity rate becomes arbitrarily fast, the optimal sampling complexity rate stays uniformly bounded. In this work, we extend the theory-to-practice gap result for learning in finite dimensions to the infinite-dimensional setting of operator learning.