I tried with both the person's name and their email. I have the role of Admin in the page and there's one other Admin. Is there a limit on the number of people with roles or some other reason I can't add a person to the page? What else could I look for?
There is no limit on the number of people with roles, the person you have selected for the Page Role must have Liked the page. If someone has not liked the page you can not give him/her page role.
Once the person like the page, you can use his/her name or username or email address to select them for Page Role.