Can an academy also be a boarding school at the same time?

From what I'm getting, an academy is for smarter students while a boarding school provides a place to live for its students. Can you have an academy boarding school? What would you call it? XYZ Academy? XYZ Boarding School? XYZ Boarding Academy? XYZ High school? XYZ Boarding School for the Gifted/Talented? (doing this for writing)