The paper weighs the applicability of Conceptual Metaphor Theory (Lakoff & Johnson 1980) and treats it as a theory of concepts, based on recent experimental findings. The outcome is controversial because, although it provides evidence for the psychological reality of cross-domain mappings, it cannot account for the activation of cross-domain mappings when the human mind processes conventional conceptual metaphors. The author suggests that it is necessary to distinguish between cross-domain mappings and the metaphoric structure of our concepts and to reconsider the Conceptual Metaphor Theory. The author proposes a different theory of concepts which approaches concepts as vague predicates, with vague extensions.