In this paper, we develop, analyze, and implement a new recursive method to conservatively account for non-Gaussian measurement errors with an uncertain correlation structure in Kalman filters (KFs). Under the assumptions of symmetric overbounding, the method guarantees a CDF overbound on the entire KF estimation error distribution. First, we leverage previous work on symmetric overbounding and frequency-domain overbounding to show how to transform a measurement domain CDF overbound into a position domain overbound. The second part of the paper evaluates the proposed method through Monte Carlo simulation for a GPS-based position estimation problem. Specifically, we show that while frequency domain overbounding produces a position domain overbound for Gaussian noise with an uncertain correlation structure, combination with symmetric overbounding is required to ensure position domain overbounding for non-Gaussian correlated noise.