We address the problem of timing-based localization in wireless networks, when an unknown fraction of data is corrupted by nonideal signal conditions. While timing-based techniques enable accurate localization, they are also sensitive to such corrupted data. We develop a robust method that is applicable to a range of localization techniques, including time-of-arrival, time-difference-of-arrival and time-difference in schedule-based transmissions. The method is nonparametric and requires only an upper bound on the fraction of corrupted data, thus obviating distributional assumptions of the corrupting noise distribution. The robustness of the method is demonstrated in numerical experiments.