In contrast to a previous study we conducted and other evidence, a recent study found no significant effects of Medicare coverage after age 65 on overall health for previously uninsured adults and significant adverse effects on survival for some of these adults. We discuss explanations for these inconsistent findings, particularly the different ways in which deaths were handled, a key methodological challenge in longitudinal analyses of health. We demonstrate that analytic approaches suitable for examining effects of coverage on health measures may not be suitable for effects on mortality. Thus, estimates may be misleading when these different outcomes are jointly modeled. We also present new survival analyses that suggest Medicare coverage significantly attenuated the rising risk of death for previously uninsured adults.