They need the data-source Connect role as well.
In Tableau Server, go to the Content page, Data Sources, and check the box next to the data source you are interested in. Click the now-highlighted Permissions link at the top of the page. Click the "edit" link in the row you want to alter, most likely the "All Users" User/Group. Make sure the Connect permission (it's not a role!) and the View permission is set to "Allow":