Large language models (LLMs) have performed well on several reasoning benchmarks, including ones that test analogical reasoning abilities. However, there is debate on the extent to which they are performing general abstract reasoning versus employing shortcuts or other non-robust processes, such as ones that overly rely on similarity to what has been seen in their training data. Here we investigate the robustness of analogy-making abilities previously claimed for one prominent class of LLMs—GPT models—on three of four domains studied by Webb et al. (2023): letter-string analogies, digit matrices, and story analogies. For each of these domains we test humans and GPT models on robustness to variants of the original analogy problems—versions that test the same abstract reasoning abilities but that are likely dissimilar from tasks in the pre-training data. The performance of a system that uses robust abstract reasoning should not decline substantially on these variants. On simple letter-string analogies, we find that while the performance of humans remains high for two types of variants we tested, the GPT models’ performance declines sharply. This pattern is less pronounced as the complexity of these analogy problems is increased, as both humans and GPT models perform poorly on both the original and variant problems requiring more complex analogies. On digit-matrix problems, we find a similar pattern but only on one out of the two types of variants we tested. Lastly, we assess the robustness of humans and GPT models on story-based analogy problems, finding that, unlike humans, the performance of GPT models are susceptible to answer-order effects, and that GPT models also may be more sensitive than humans to paraphrasing. This work provides evidence that, despite previously reported successes of GPT models on zero-shot analogical reasoning, these models often lack the robustness of zero-shot human analogy-making, exhibiting brittleness on most of the variations we tested. More generally, this work points to the importance of carefully evaluating AI systems not only for accuracy but also robustness when testing their cognitive capabilities.