I do not believe that is true. We definitely see elves using magic in the books, such as the magic used by Elrond and Galadriel to protect Rivendell and Lothlorien – sure, the Rings are part of that, but not the entiriety. Boromir and Faramir had prophetic dreams. Beorn (and his kin?) can change shape. IIRC, many of the Nazgûl were also some kind of magic-users in their mortal lives – there's a reason the Witch-King of Angmar is called that. And the Blue Wizards were said to have started magic traditions of their own in the East, though we don't know much about those.