According to much of theoretical linguistics, a fair amount of our linguistic knowledge is innate. One of the best-known (and most contested) kinds of evidence for a large innate endowment is the argument from the poverty of the stimulus (APS). An APS obtains when human learners systematically make inductive leaps that are not warranted by the linguistic evidence. A weakness of the APS has been that it is very hard to assess what is warranted by the linguistic evidence. Current artificial neural networks appear to offer a handle on this challenge, and a growing literature has started to explore the potential implications of such models to questions of innateness. We focus on Wilcox, Futrell, and Levy’s (2024) use of several different networks to examine the available evidence as it pertains to wh -movement, including island constraints. WFL conclude that the (presumably linguistically neutral) networks acquire an adequate knowledge of wh -movement, thus undermining an APS in this domain. We examine the evidence further, looking in particular at parasitic gaps and across-the-board movement, and argue that current networks do not succeed in acquiring or even adequately approximating wh -movement from training corpora roughly the size of the linguistic input that children receive. We also show that the performance of one of the models improves considerably when the training data are artificially enriched with instances of parasitic gaps and across-the-board movement. This finding suggests, albeit tentatively, that the networks’ failure when trained on natural, unenriched corpora is due to the insufficient richness of the linguistic input, thus supporting the APS.