Clinical notes are becoming an increasingly important data source for machine learning (ML) applications in healthcare. Prior research has shown that deploying ML models can perpetuate existing biases against racial minorities, as bias can be implicitly embedded in data. In this study, we investigate the level of implicit race information available to ML models and human experts and the implications of model-detectable differences in clinical notes. Our work makes three key contributions. First, we find that models can identify patient self-reported race from clinical notes even when the notes are stripped of explicit indicators of race. Second, we determine that human experts are not able to accurately predict patient race from the same redacted clinical notes. Finally, we demonstrate the potential harm of this implicit information in a simulation study, and show that models trained on these race-redacted clinical notes can still perpetuate existing biases in clinical treatment decisions.