In this paper, we provide a mathematical framework for the generation and application of contingency tables with bounds in situations where obtaining exact frequency distributions is not possible. We focus on the Integrated Clinical and Environmental Exposures Service (ICEES). ICEES is an open service that provides access to sensitive clinical data that have been integrated with public exposures data. The concept of bounded contingency tables is motivated by ICEES’ privacy restrictions, which prohibit the release of electronic health record data on cohorts of fewer than 10 patients. While this service has unique limitations,the concept of bounded contingency tables is easily generalizable, and has been previously explored by others in the context of privacy restrictions imposed on open clinical data.