To collaborate effectively with humans, language models must be able to explain their decisions in natural language. We study a specific type of self-explanation: self-generated counterfactual explanations (SCEs), where a model explains its prediction by modifying the input such that it would have predicted a different outcome. We evaluate whether LLMs can produce SCEs that are valid, achieving the intended outcome, and minimal, modifying the input no more than necessary. When asked to generate counterfactuals, we find that LLMs typically produce SCEs that are valid, but far from minimal, offering little insight into their decision-making behaviour. Worryingly, when asked to generate minimal counterfactuals, LLMs typically make excessively small edits that fail to change predictions. The observed validity-minimality trade-off is consistent across several LLMs, datasets, and evaluation settings. Our findings suggest that SCEs are, at best, an ineffective explainability tool and, at worst, can provide misleading insights into model behaviour. Proposals to deploy LLMs in high-stakes settings must consider the impact of unreliable self-explanations on downstream decision-making. Our code is available at https://github.com/HarryMayne/SCEs.