Concept bottleneck models predict outcomes from high-level concepts detected in inputs. Although concepts provide a simple way to reap benefits from interpretability, very few datasets include concept labels. This limits researchers' ability to determine which problems are suitable for these models, isolate the factors that drive their performance or lead to failures, or uncover which algorithms perform well. In this paper, we develop synthetic benchmarks for concept-bottleneck models, focusing on their two main use cases: decision support, in which models assist humans in making better decisions, and automation, in which models handle routine tasks without supervision. Our benchmarks can generate labeled datasets while controlling for properties that affect performance, including data modality, concept choice, annotation quality, and completeness. We demonstrate how the benchmarks can be used to evaluate representative classes of concept bottleneck models. Our demonstrations show how the benchmarks can diagnose failure modes and guide follow-up testing.