Model errors are increasingly seen as a fundamental performance limiter in both Numerical Weather Prediction and Climate Prediction simulations run with state‐of‐the‐art Earth‐system digital twins. This has motivated recent efforts aimed at estimating and correcting the systematic, predictable components of model error in a consistent data assimilation framework. While encouraging results have been obtained with a careful examination of the spatial aspects of the model error estimates, less attention has been devoted to the time correlation aspects of model errors and their impact on the assimilation cycle. In this work we employ a Lagged Analysis Increment Covariance (LAIC) diagnostic to gain insight in the temporal evolution of systematic model errors in the ECMWF operational data assimilation system, evaluate the effectiveness of the current weak constraint 4D‐Var algorithm in reducing these types of errors and, based on these findings, start exploring new ideas for the development of model error estimation and correction strategies in data assimilation.