Conventional wisdom holds that interleaved pooling layers in convolutional neural networks lead to stability to small translations and deformations. In this work, we investigate this claim empirically. We find that while pooling confers stability to deformation at initialization, the deformation stability at each layer changes significantly over the course of training and even decreases in some layers, suggesting that deformation stability is not unilaterally helpful. Surprisingly, after training, the pattern of deformation stability across layers is largely independent of whether or not pooling was present. We then show that a significant factor in determining deformation stability is filter smoothness. Moreover, filter smoothness and deformation stability are not simply a consequence of the distribution of input images, but depend crucially on the joint distribution of images and labels. This work demonstrates a way in which biases such as deformation stability can in fact be learned and provides an example of understanding how a simple property of learned network weights contributes to the overall network computation.