In this paper, an introduction to some existing Item Response Theory (IRT) models for Local Item Dependencies (LIDs) is presented together with an application. First, four LID models introduced by Hoskens and De Boeck (1997) are discussed and a more general model is derived. Second, it is described in broad outline how these models can be estimated and tested statistically via a loglinear approach. Third, responses are examined on questions that cannot be understood by the respondents. For this purpose a Polish Questionnaire was used in which there was no clear indication in the items of which response (yes or no) had to be chosen. An interpretable dependency structure was found for the first part of the test, but it could not be validated for the entire test. In conclusion, the problems are discussed that are inherent to the classical estimation and testing procedure when dealing with local item dependencies and some solutions are offered.