Over the past decade, the Tunable Laser Spectrometer (TLS) on NASA's Curiosity rover has reported several detections of methane on Mars, attracting attention due to the potential astrobiological implications of its presence. Here, we re-analyze published TLS data, identifying issues in data robustness and reduction. We find that the TLS foreoptics chamber typically contained methane abundances that were 3–4 orders of magnitude greater than those reported in the sample cell, alongside unexpected and rapidly varying pressure changes inside the instrument. Using information from unreported experiments where methane diffusion into the cell was observed, we estimate a gas transport coefficient and develop a model to simulate gas exchanges between the two compartments in typical experiments, investigating the implications for methane measurements. We find that tiny leaks (<0.1\% of foreoptics methane) would suffice to explain the reported atmospheric methane measurements—leaks that are otherwise undetectable from housekeeping data. Furthermore, in an analysis of five experiments where more complete data are available, we find that the TLS retrieval method—which averages discrepant methane levels from the three lines of the R3 triplet as if the three lines were independent, rather than fitting the spectrum to the distinctive pattern of the triplet itself—likely underestimates uncertainties. The probability that three methane levels from individual triplet lines are disparate in all five experiments is typically ∼10−3, suggesting the presence of systematic errors that are unaccounted for in previously reported methane levels. Finally, we propose a constructive two-step experiment to further investigate our findings.