Transformer interpretability aims to understand the algorithm implemented by a learned Transformer by examining various aspects of the model, such as the weight matrices or the attention patterns.In this work, through a combination of theoretical results and carefully controlled experiments on synthetic data, we take a critical viewof methods that exclusively focus on individual parts of the model, rather than consider the network as a whole.We consider a simple synthetic setup of learning a (bounded) Dyck language. Theoretically, we show that the set of models that (exactly or approximately) solve this task satisfy a structural characterization derived from ideas in formal languages (the pumping lemma).We use this characterization to show that the set of optima is qualitatively rich; in particular, the attention pattern of a single layer can be "nearly randomized", while preserving the functionality of the network.We also show via extensive experiments that these constructions are not merely a theoretical artifact: even with severe constraints to the architecture of the model, vastly different solutions can be reached via standard training. Thus, interpretability claims based on inspecting individual heads or weight matrices in the Transformer can be misleading.