AI is popping up everywhere, in all sorts of forms, claiming to be able to do almost everything. Caught between fantasy and reality, AI is still largely excluded from health services and establishments otherwise welcoming to innovation. This study, conducted simultaneously in France and Quebec, shows how the logic natural to AI fails to fit with the norms and bases of the healthcare sector, revealing as well how little has been done by the relevant actors to remedy this. The gap observed has less to do with AI's potential as a tool than with the rules of how the medical world, and especially the hospital system, work.