I didn't yet find a post about this anywhere so adding this here in case anyone else encounters the same problem.
If a user has an Excel with a formula that references a cube and a scenario that is not valid for that cube (i.e. there is no CubeRoot profile for that cube/scenario combination), all cells in the Excel (not just the cell with the mistake) will show #REFRESH instead of the expected values. This will somehow corrupt the Excel so that the problem continues even after the cell is removed and the sheet is refreshed! The only way to get all the other cells to retrieve values correctly again seems to be to remove the cell with the mistake, restart Excel, and then refresh again.