In this paper, we seek to measure how much information a component in a neural network could extract from the representations fed into it. Our work stands in contrast to prior probing work, most of which investigates how much information a model’s representations contain. This shift in perspective leads us to propose a new principle for probing, the architectural bottleneck principle: In order to estimate how much information a given component could extract, a probe should look exactly like the component. Relying on this principle, we estimate how much syntactic information is available to transformers through our attentional probe, a probe that exactly resembles a transformer’s self-attention head. Experimentally, we find that, in three models (BERT, ALBERT, and RoBERTa), a sentence’s syntax tree is mostly extractable by our probe, suggesting these models have access to syntactic information while composing their contextual representations. Whether this information is actually used by these models, however, remains an open question.