We study the neural network (NN) compression problem, viewing the tension between the compression ratio and NN performance through the lens of rate-distortion theory. We choose a distortion metric that reflects the effect of NN compression on the model output and then derive the tradeoff between rate (compression ratio) and distortion. In addition to characterizing theoretical limits of NN compression, this formulation shows that pruning, implicitly or explicitly, must be a part of a good compression algorithm. This observation bridges a gap between parts of the literature pertaining to NN and data compression, respectively, providing insight into the empirical success of pruning for NN compression. Finally, we propose a novel pruning strategy derived from our information-theoretic formulation and show that it outperforms the relevant baselines on CIFAR-10 and ImageNet datasets.