Estimators of common parameters are often argued to be robust to random heterogeneity. One example of a situation in which such arguments have been made is when estimating structural breaks in panel data. In a seminal paper, Bai (Common Breaks in Means and Variances for Panel Data. Journal of Econometrics 157, 78-92, 2010) considered a model with a breaking mean. Consistency of the least squares breakpoint estimator is established under the assumption of a common break; however, the estimator is claimed to be valid also under heterogenous breaks, provided that they are randomly distributed with a common mean. The present paper shows that this last claim need not be correct. One implication of this finding is that robustness to random heterogeneity does not come automatically, but has to be verified on a case-to-case basis.