Attention-based mechanisms are widely used in machine learning, most prominently in transformers. However, hyperparameters such as the number of attention heads and the attention rank (i.e., the query/key dimension) are set nearly the same way in all realizations of this architecture, without theoretical justification. In this paper, we prove that the rank can have a dramatic effect on the representational capacity of attention. This effect persists even when the number of heads and the parameter count are very large. Specifically, we present a simple and natural target function based on nearest neighbor search that can be represented using a single full-rank attention head for any sequence length, but that cannot be approximated by a low-rank attention layer even on short sequences unless the number of heads is exponential in the embedding dimension. Thus, for this target function, rank is what determines an attention layer's power. We show that, for short sequences, using multiple layers allows the target to be approximated by low-rank attention; for long sequences, we conjecture that full-rank attention is necessary regardless of depth. Finally, we present experiments with standard multilayer transformers that validate our theoretical findings. They demonstrate that, because of how standard transformer implementations set the rank, increasing the number of attention heads can severely decrease accuracy on certain tasks.