Assessing competencies of students with special educational needs in learning (SEN-L) poses a challenge for large-scale assessments (LSAs). For students with SEN-L, the available competence tests may fail to yield test scores of high psychometric quality, which are—at the same time—measurement invariant to test scores of general education students. We investigated whether we can identify a subgroup of students with SEN-L, for which measurement invariant competence measures of adequate psychometric quality may be obtained with tests available in LSAs. We furthermore investigated whether differences in test-taking behavior may explain dissatisfying psychometric properties and measurement non-invariance of test scores within LSAs. We relied on person fit indices and mixture distribution models to identify students with SEN-L for whom test scores with satisfactory psychometric properties and measurement invariance may be obtained. We also captured differences in test-taking behavior related to guessing and missing responses. As a result we identified a subgroup of students with SEN-L for whom competence scores of adequate psychometric quality that are measurement invariant to those of general education students were obtained. Concerning test taking behavior, there was a small number of students who unsystematically picked response options. Removing these students from the sample slightly improved item fit. Furthermore, two different patterns of missing responses were identified that explain to some extent problems in the assessments of students with SEN-L.