We give a classification of the complexity of DL-Lite logics extended with role inclusion axioms. We show that the data complexity of instance checking becomes P-hard in the presence of functionality constraints, and coNP-hard if arbitrary number restrictions are allowed, even with primitive concept inclusions. The combined complexity of satisfiability in this case jumps to ExpTime. On the other side, the combined complexity for the logics without number restrictions depends only on the form of concept inclusions and can range from NLogSpace and P to NP; the data complexity for such logics stays in LogSpace.