nomadic yeti wrote:
.. is the email account you're trying to add also the same email address for your Apple ID? I just had the same issue. I had to completely logout of iCloud on the Mac (by going through the "Internet Accounts" preference pane of Settings) before I could add the email account that "already existed". I was able to add the account then, but I wasn't able to log back in to iCloud after that (it said it was already logged in even though no iCloud account showed up on in the preference pane). I then had to leave the "Internet Accounts" preference pane and click on the iCloud preference pane in Settings. Sure enough I was still logged into my iCloud account there. I just had to click on all the programs that I wanted to use iCloud. Everything worked fine after that.
Thanks @nomadic yeti, that worked for me with an Internode IMAP email account, that failed to log in when I upgraded to El Capitan. I would repeatedly enter the password to no avail. So I deleted the mail account and tried to set it up from new, couldn't do this as I was getting "This account already exists" lies. And yes, that email address is my iCloud/Apple ID email.
I was able to log back in to iCloud in the Internet Accounts pane however, unlike you. So for me the whole process was:
1) quit Mail app
2) System Preferences > Internet Accounts > select iCloud account “-“ key to delete account. (say yes to all the warnings)
3) start Mail app, Preferences > Accounts > “+” button
4) enter email account info, account gets created in Mail. Quit Mail app
5) System Preferences > Internet Accounts > “+” to add your iCloud account (tick boxes for which services you wish to have synced).