Electronic medical records (EMRs) data is increasingly incorporated into genome-phenome association studies. Investigators hope to share data, but there are concerns it may be "re-identified" through the exploitation of various features, such as combinations of standardized clinical codes. Formal anonymization algorithms (e.g., k-anonymization) can prevent such violations, but prior studies suggest that the size of the population available for anonymization may influence the utility of the resulting data. We systematically investigate this issue using a large-scale biorepository and EMR system through which we evaluate the ability of researchers to learn from anonymized data for genome-phenome association studies under various conditions.