We attempt to investigate the effects of using residuals from robust regression replacing OLS residuals in test statistics for the normality of the errors. We have found that this can lead to substantially improved ability to detect lack of normality in suitable situations. We derive the asymptotic distribution of the robustified normality test as chi-squared with 2 degrees of freedom under the null hypothesis of normality of the error terms. The high breakdown property of the test statistic is discussed. By using simulations, we have found that situations where a small subpopulation exhibits characteristics which are different from the main population are the ones which ideally suit to the use of robustified normality tests. We have employed several real data sets from the literature to show that these types of situations arise frequently in real data sets.