How should we compare the capabilities of language models and humans? Here, I consider a case study: processing of recursively nested grammatical structures. Prior work has suggested that language models cannot handle these structures as reliably as humans can. However, the humans were provided with instructions and training before being evaluated, while the language models were evaluated zero-shot. I therefore attempt to more closely match the evaluation paradigms by providing language models with few-shot prompts. A simple prompt, which contains substantially less content than the human training, allows large language models to consistently outperform the human results. The same prompt even allows extrapolation to more deeply nested conditions than have been tested in humans. Further, a reanalysis of the prior human experiments suggests that the humans may not perform above chance at the difficult structures initially. These results suggest that large language models can in fact process recursively nested grammatical structures comparably to humans. This case study highlights how discrepancies in the quantity of experiment-specific context can confound comparisons of language models and humans. I use this case study to reflect on the broader challenge of comparing human and model capabilities, and to suggest that there is an important difference between evaluating cognitive models of a specific phenomenon and evaluating broadly-trained models.