Register data that originate from administrative or other secondary sources are increasingly being used to generate statistical outputs directly. The coverage of the input datasets is an important issue in this respect. Traditionally capture-recapture models have been used to deal with multiple list enumerations subjected to undercoverage errors. The aim of this article is to scope possible approaches to modelling capture-recapture data with additional overcoverage error. Attention is primarily given to model interpretations and conditions under which a model may provide a plausible basis for estimation and uncertainty evaluation. The setting with two list enumerations is examined in depth as it is the most common in practice. Models that can be extended to include more than two lists are identified. An additional independent coverage survey with only undercoverage error is always needed for estimation. Potential application to census coverage-error adjustment is discussed.