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For several decades, the resampling based bootstrap has been widely used for computing confidence intervals (CIs) for applications where no exact method is available. However, there are many applications where the resampling bootstrap method cannot be used. These include situations where the data are heavily censored due to the success response being a rare event, situations where there is insufficient mixing of successes and failures across the explanatory variable(s), and designed experiments where the number of parameters is close to the number of observations. These three situations all have in common that there may be a substantial proportion of the resamples where it is not possible to estimate all of the parameters in the model. This article reviews the fractional-random-weight bootstrap method and demonstrates how it can be used to avoid these problems and construct CIs in a way that is accessible to statistical practitioners. The fractional-random-weight bootstrap method is easy to use and has advantages over the resampling method in many challenging applications.