The linear logistic test model (LLTM) is a valuable and approved tool in educational research, as it allows for modelling cognitive components involved in a cognitive task. It allows for a rigorous assessment of fit by means of a Likelihood Ratio Test (LRT). This approach is genuine to the Rasch family of models, yet it suffers from the unsolved problem of optimal sample size determination. The study presents a simulation study showing that comparably small deviations of item parameters can be detected with large power. However, even for models that have been rejected by the LRT, person parameters of the Rasch model (RM) and those of the non-fitting RM are surprisingly similar. This result suggests a reconsideration of the testing criteria when person parameter estimates are of major concern, as the LRT might prove overly sensitive for practical applications. (Contains 3 tables and 3 figures.)