A general formula for computing the breakdown point in robustness for the $t$th bootstrap quantile of a statistic $T_n$ is obtained. The answer depends on $t$ and the breakdown point of $T_n$. Since the bootstrap quantiles are vital ingredients of bootstrap confidence intervals, the theory has implications pertaining to robustness of bootstrap confidence intervals. For certain $L$ and $M$ estimators, a robustification of bootstrap is suggested via the notion of Winsorization.
"Breakdown theory for bootstrap quantiles." Ann. Statist. 26 (5) 1719 - 1732, October 1998. https://doi.org/10.1214/aos/1024691354