The standard machinery for system identification of linear time-invariant (LTI) models delivers a nominal model and a confidence (uncertainty) region around it, based on (second order moment) residual analysis and covariance estimation. In most cases this gives an uncertainty region that tends to zero as more and more data become available, even if the true system is nonlinear and/or time-varying. In this paper, the reasons for this are displayed, and a characterization of the limit LTI model is gh,en under quite general conditions. Various ways are discussed, and tested, to obtain a more realistic limiting model, with uncertainty. These should reflect the distance to the true possibly nonlinear, time-varying system, and also form a reliable basis for robust LTI control design.