In some randomized (drug versus placebo) clinical trials, the estimand of interest is the between-treatment difference in population means of a clinical endpoint that is free from the confounding effects of "rescue" medication (e.g., HbA1c change from baseline at 24 weeks that would be observed without rescue medication regardless of whether or when the assigned treatment was discontinued). In such settings, a missing data problem arises if some patients prematurely discontinue from the trial or initiate rescue medication while in the trial, the latter necessitating the discarding of post-rescue data. We caution that the commonly used mixed-effects model repeated measures analysis with the embedded missing at random assumption can deliver an exaggerated estimate of the aforementioned estimand of interest. This happens, in part, due to implicit imputation of an overly optimistic mean for "dropouts" (i.e., patients with missing endpoint data of interest) in the drug arm. We propose an alternative approach in which the missing mean for the drug arm dropouts is explicitly replaced with either the estimated mean of the entire endpoint distribution under placebo (primary analysis) or a sequence of increasingly more conservative means within a tipping point framework (sensitivity analysis); patient-level imputation is not required. A supplemental "dropout = failure" analysis is considered in which a common poor outcome is imputed for all dropouts followed by a between-treatment comparison using quantile regression. All analyses address the same estimand and can adjust for baseline covariates. Three examples and simulation results are used to support our recommendations.