Work using artificial languages as training input has shown that LSTMs are capable of inducing the stack-like data structures required to represent context-free and certain mildly context-sensitive languages — formal language classes which correspond in theory to the hierarchical structures of natural language. Here we present a suite of experiments probing whether neural language models trained on linguistic data induce these stack-like data structures and deploy them while incrementally predicting words. We study two natural language phenomena: center embedding sentences and syntactic island constraints on the filler–gap dependency. In order to properly predict words in these structures, a model must be able to temporarily suppress certain expectations and then recover those expectations later, essentially pushing and popping these expectations on a stack. Our results provide evidence that models can successfully suppress and recover expectations in many cases, but do not fully recover their previous grammatical state.