Deep learning training accesses vast amounts of data at high velocity, posing challenges for datasets retrieved over commodity networks and storage devices. We introduce a way to dynamically reduce the overhead of fetching and transporting training data with a method we term Progressive Compressed Records (PCRs). PCRs deviate from previous formats by using progressive compression to convert a single dataset into multiple datasets of increasing fidelity---all without adding to the total dataset size. Empirically, we implement PCRs and evaluate them on a wide range of datasets: ImageNet, HAM10000, Stanford Cars, and CelebA-HQ. Our results show that different tasks can tolerate different levels of compression. PCRs use an on-disk layout that enables applications to efficiently and dynamically access appropriate levels of compression at runtime. In turn, we demonstrate that PCRs can seamlessly enable a 2x speedup in training time on average over baseline formats.