It is widely believed that natural image data exhibits low-dimensional structure despite being embedded in a high-dimensional pixel space. This idea underlies a common intuition for the success of deep learning and has been exploited for enhanced regularization and adversarial robustness. In this work, we apply dimension estimation tools to popular datasets and investigate the role of low dimensional structure in neural network learning. We find that common natural image datasets indeed have very low intrinsic dimension relative to the high number of pixels in the images. Additionally, we find that low dimensional datasets are easier for neural networks to learn. We validate our dimension estimation tools synthetic data generated by GANs in which we can manipulate intrinsic dimension.