This paper concerns the analysis of data for cases where there are NO occurrences of "failures" when there is reason to assume a Weibull distribution with known slope parameter for the characteristic being studied. All samples survive a time t in an unfailed condition. In a series of independent tests conducted under similar conditions, if we observe no failures by time t we call this a "success" run of length n. In this paper the classical non-parametric success run is first reviewed and it is shown what can be said about the relationship among reliability, confidence and sample size. Next, the Weibull distribution is explored for modeling the success run test when it is reasonable to assume a value for the Weibull slope parameter, b. We discuss under what conditions it is reasonable to assume a Weibull slope and derive a variety of relationships among sample size, confidence, reliability, Weibull characteristic life and test length that prove useful for test planning and data analysis when the Weibull slope can be approximated.