Concept-based interpretability methods aim to explain a deep neural network model’s components and predictions using a pre-defined set of semantic concepts. These methods evaluate a trained model on a new, “probe” dataset and correlate the model’s outputs with concepts labeled in that dataset. Despite their popularity, they suffer from limitations that are not well-understood and articulated in the literature. In this work, we identify and analyze three commonly overlooked factors in concept-based explanations. First, we find that the choice of the probe dataset has a profound impact on the generated explanations. Our analysis reveals that different probe datasets lead to very different explanations, suggesting that the generated explanations are not generalizable outside the probe dataset. Second, we find that concepts in the probe dataset are often harder to learn than the target classes they are used to explain, calling into question the correctness of the explanations. We argue that only easily learnable concepts should be used in concept-based explanations. Finally, while existing methods use hundreds or even thousands of concepts, our human studies reveal a much stricter upper bound of 32 concepts or less, beyond which the explanations are much less practically useful. We discuss the implications of our findings and provide suggestions for future development of concept-based interpretability methods. Code for our analysis and user interface can be found at https://github.com/princetonvisualai/OverlookedFactors.