This paper considers estimation of truncated and censored regression models with fixed effects in panel data. Up until now, no estimator has been shown to be consistent as the cross section dimension increases with the time dimension fixed. Trimmed least absolute deviations (LAD) and trimmed least squares estimators are proposed for the case where the panel is of length two, and it is proven that they are consistent and asymptotically normal under suitable regularity conditions. It is not necessary to maintain parametric assumptions on the error terms to obtain this result. Because three of the four estimators are defined as minimizers of nondifferentiable functions, traditional methods cannot be used to establish asymptotic normality. Instead, the approach of Pakes and Pollard (1989) is used. A small scale Monte Carlo study demonstrates that these estimators can perform well in small samples. Despite their nonlinear nature, the estimators are easy to calculate in practice, as are consistent estimators of their asymptotic variances. Generalization of the estimators to panels of arbitrary length is briefly discussed.