Summary: One aim of robust regression is to find estimators with high finite sample breakdown points. Although various robust estimators have been proposed in logistic regression models, their breakdown points are not yet known. Here it is shown for logistic regression models with binary data that there is no estimator with a high finite sample breakdown point, provided the estimator has to fulfill a weak condition. In logistic regression models with large strata however, a modification of P. J. Rousseeuw’s [J. Am. Stat. Assoc. 79, 871-880 (1984; Zbl 0547.62046)] least median of square estimator is shown to have a finite sample breakdown point of approximately \(1/2\).