We studied the learnability of English filler-gap dependencies and the “island” constraints on them by assessing the generalizations made by autoregressive (incremental) language models that use deep learning to predict the next word given preceding context. Using factorial tests inspired by experimental psycholinguistics, we found that models acquire not only the basic contingency between fillers and gaps, but also the unboundedness and hierarchical constraints implicated in the dependency. We evaluated a model’s acquisition of island constraints by demonstrating that its expectation for a filler-gap contingency is attenuated within an island environment. Our results provide empirical evidence against the argument from the poverty of the stimulus for this particular structure.