There exist modal logics that are validated by their canonical frames but are not sound and complete for any elementary class of frames. Continuum many such bimodal logics are exhibited, including one of each degree of unsolvability, and all with the finite model property. Monomodal examples are also constructed that extend K4 and are related to the proof of non-canonicity of the McKinsey axiom. We dedicate this paper to Max Cresswell, a pioneer in the study of canonicity, on the occasion of his 65th birthday.