We consider the Roe-Woodroofe construction of confidence intervals for the case of a Poisson distributed variate, where the mean is the sum of a known background and an unknown non-negative signal. The RW construction is based on a conditional pdf and posseses conditional coverage such that the intervals contain the true value of the signal with prescribed probability only for subsets of the full set of repeated experiments. The intervals do not have coverage in the usual sense but can be made to have such with a modification that does not affect the believability and other desirable features of this attractive construction. A similar modification can be used to provide coverage to a recently discussed application of this method to the gaussian-with-boundary problem. In both cases the result is equivalent to a construction in which the defining contour is the bayesian upper limit (uniform prior) for observations near the physical bound and the bayesian (or frequentist) central confidence interval asymptotically.