Deep neural networks (DNNs) have achieved state-of-the-art performance in identifying gene regulatory sequences, but they have provided limited insight into the biology of regulatory elements due to the difficulty of interpreting the complex features they learn. Several models of how combinatorial binding of transcription factors, i.e. the regulatory grammar, drives enhancer activity have been proposed, ranging from the flexible TF billboard model to the stringent enhanceosome model. However, there is limited knowledge of the prevalence of these (or other) sequence architectures across enhancers. Here we perform several hypothesis-driven analyses to explore the ability of DNNs to learn the regulatory grammar of enhancers. We created synthetic datasets based on existing hypotheses about combinatorial transcription factor binding site (TFBS) patterns, including homotypic clusters, heterotypic clusters, and enhanceosomes, from real TF binding motifs from diverse TF families. We then trained deep residual neural networks (ResNets) to model the sequences under a range of scenarios that reflect real-world multi-label regulatory sequence prediction tasks. We developed a gradient-based unsupervised clustering method to extract the patterns learned by the ResNet models. We demonstrated that simulated regulatory grammars are best learned in the penultimate layer of the ResNets, and the proposed method can accurately retrieve the regulatory grammar even when there is heterogeneity in the enhancer categories and a large fraction of TFBS outside of the regulatory grammar. However, we also identify common scenarios where ResNets fail to learn simulated regulatory grammars. Finally, we applied the proposed method to mouse developmental enhancers and were able to identify the components of a known heterotypic TF cluster. Our results provide a framework for interpreting the regulatory rules learned by ResNets, and they demonstrate that the ability and efficiency of ResNets in learning the regulatory grammar depends on the nature of the prediction task.