Regarding two accounts on one battle.net acct

I apologize in advance if this is something that should be painfully obvious, but I have two different WoW accounts both linked to the same battle.net account and I was wondering if that fact affects the ability to play both accounts simultaneously.

What I'm getting at is that I want to play on one account while my son plays on the other, does having the same login information for both (with the only difference being which account is selected on the drop-down menu) prevent us from doing that?

I would test this myself but only one of those accounts is currently active and I wouldn't want to waste $15 on an account I personally wouldn't have any use for.