The Advanced Receiver Autonomous Integrity Monitoring (ARAIM) concept relies on the characterization of conservative error bounds of the clock and ephemeris nominal errors to achieve the required level of integrity. . In this paper, we attempt to investigate the uncertainty inherent in estimating the Gaussian bounding parameters. If the past nominal error bounding parameters are stable, then we can have some confidence that they will stay stable in the future, a conclusion critical for GNSS integrity analysis. We used an error bounding algorithm to examine the GPS and Galileo satellite clock and ephemeris error bounding behavior for the past 12 years. We found that the error distribution overbounding parameter estimation has increased stability after removing the near-fault data points. We evaluated the uncertainty in the bounding process using both the bootstrap and the training-validation methods. We found that a larger data set size significantly reduces the uncertainty, and using 12 years for GPS appears to be enough to characterize the bounding parameter behavior due to its stability, whereas the two years of Galileo’s available data set is not.