Empirical economists are often deterred from the application of nonlinear fixed effects models mainly for two reasons: the incidental parameter problem and the computational challenge even in moderately large panels. Using the example of binary choice models with individual and time fixed effects, we show how both issues can be alleviated by combining the asymptotic bias corrections of Fernández-Val and Weidner (2016) with an estimation algorithm proposed by Stammann (2018). We conduct extensive simulation experiments to analyze the statistical properties of various (bias-corrected) estimators in large and even unbalanced panels. Our simulation results provide new insights that are relevant for the application of bias corrections in empirical work. Overall, we find that analytically bias corrected estimators are clearly preferable, especially for unbalanced panels.