Abstract A robust system for ontology-based data access should provide meaningful answers to queries even when the data conflicts with the ontology. This can be accomplished by adopting an inconsistencytolerant semantics, with the consistent query answering (CQA) semantics being the most prominent example. Unfortunately, query answering under the CQA semantics has been shown to be computationally intractable, even when extremely simple ontology languages are considered. In this paper, we address this problem by proposing two new families of inconsistency-tolerant semantics which approximate the CQA semantics from above and from below and converge to it in the limit. We study the data complexity of conjunctive query answering under these new semantics, and show a general tractability result for all known first-order rewritable ontology languages. We also analyze the combined complexity of query answering for ontology languages of the DL-Lite family.