Two general principles have played a crucial role in the recent debate on concepts. On the one hand, we want to allow different subjects to have the same concepts, thus accounting for concept publicity: concepts are ‘the sort of thing that people can, and do, share’. On the other hand, a subject who finds herself in a so-called ‘Frege case’ appears to have different concepts for the same object: for instance, Lois Lane has two distinct concepts SUPERMAN and CLARK KENT which refer to the same person. Several theories have tried to meet both of these constraints at the same time. But should we really try to satisfy both principles? This paper will argue that the traditional project of fulfilling these two constraints has been a misguided one. Through a variation on classic identity mistake cases, I will show that our two desiderata are inconsistent: it would thus be impossible to incorporate both of them in our best theory of concepts.