The dedicated memory of hardware accelerators can be insufficient to store all weights and/or intermediate states of large deep learning models. Although model parallelism is a viable approach to reduce the memory pressure issue, significant modification of the source code and considerations for algorithms are required. An alternative solution is to use out-of-core methods instead of, or in addition to, data parallelism. We propose a performance model based on the concurrency analysis of out-of-core training behavior, and derive a strategy that combines layer swapping and redundant recomputing. We achieve an average of 1.52x speedup in six different models over the state-of-the-art out-of-core methods. We also introduce the first method to solve the challenging problem of out-of-core multi-node training by carefully pipelining gradient exchanges and performing the parameter updates on the host. Our data parallel out-of-core solution can outperform complex hybrid model parallelism in training large models, e.g. Megatron-LM and Turning-NLG.