The authors study the application of the bootstrap to a class of estimators which converge at a nonstandard rate to a nonstandard asymptotic distribution. They provide a theoretical framework to study its asymptotic behaviour. A simulation study clearly shows that in the case of an estimator such as Chernoff’s estimator of the mode (1964) usually the basic bootstrap confidence intervals drastically undercover while the percentile bootstrap intervals overcover. This is one of the rare instances where basic and percentile confidence intervals, which have exactly the same length, behave in a very different way. In the case of Chernoff’s estimator, if the distribution is symmetric, it is possible to bootstrap from a smooth symmetric estimator of the distribution for which the basic bootstrap confidence intervals will have the claimed coverage probability while the percentile bootstrap interval will have an asymptotic coverage of 1!