Several methods are available for the design of phase II clinical trials with binary endpoints. A primary assumption for most methods is that observations on the endpoint of interest are uncorrelated; however, this assumption is violated if an individual patient provides more than one observation on the endpoint of interest. In such cases, one solution is to use a summary measure for each patient; an alternative solution is to perform an observation-specific analysis using a technique that properly accounts for the correlation. In this paper, we investigate the effect that ignoring correlated observations can have on the design properties of the typical phase II clinical trial. In cases in which an observation-specific analysis is desirable, we propose a simple method that adjusts a standard one- or two-stage phase II design to account for loss of information due to correlated observations. Simulations demonstrate that the method ensures that type I and type II error rate design requirements are met even in the presence of strong correlation. We develop the method in the context of phase II oncology trials, but the method applies readily to other clinical areas in which multiple responses per patient are of interest.