Recent empirical results have sparked a debate about whether or not Large Language Models (LLMs) are capable of Theory of Mind (ToM). While some have found LLMs to be successful on ToM evaluations such as the False Belief task, others have shown that their performance is not robust against trivial alterations to stimuli. In this paper, we introduce SCALPEL---a technique to incrementally modify stimuli to test different specific hypotheses about why LLMs fail---and apply this method to the `transparent-access' modification of the unexpected contents task. Our results suggest that LLMs often do poorly because they fail to make essential common-sense inferences, such as that seeing a transparent container implies recognizing its contents. We conclude that while modern LLMs go beyond mere pattern matching, they still fall short of robust human-like ToM. We argue that SCALPEL can help cognitive scientists examine LLMs' capabilities in finer detail and provide insight into alternative mechanisms by which tasks that are used to assess human cognition might be completed.