Cloud computing provides a powerful yet low-cost environment for distributed deep learning workloads. However, training complex deep learning models often requires accessing large amounts of data, which can easily exceed the capacity of local disks. Prior research often overlooks this training data problem by implicitly assuming that data is available locally or via low latency network-based data storage. Such implicit assumptions often do not hold in a cloud-based training environment, where deep learning practitioners create and tear down dedicated GPU clusters on demand, or do not have the luxury of local storage, such as in serverless workloads. In this work, we investigate the performance of distributed training that leverages training data residing entirely inside cloud storage buckets. These buckets promise low storage costs, but come with inherent bandwidth limitations that make them seem unsuitable for an efficient training solution. To account for these bandwidth limitations, we propose the use of two classical techniques, namely caching and pre-fetching, to mitigate the training performance degradation. We implement a prototype, DELI, based on the popular deep learning framework PyTorch by building on its data loading abstractions. We then evaluate the training performance of two deep learning workloads using Google Cloud's NVIDIA K80 GPU servers and show that we can reduce the time that the training loop is waiting for data by 85.6%-93.5% compared to loading directly from a storage bucket - thus achieving comparable performance to loading data directly from disk - while only storing a fraction of the data locally at a time. In addition, DELI has the potential of lowering the cost of running a training workload, especially on models with long per-epoch training times.