AH! Here is what happens, when you log in with your iPhone your user and password are tracked by the Bluesocket router at your institution. There is no way to "log out" it just takes 15 minutes of inactivity and the BlueSocket router disconnects you and removes your IP Address and user from the routing table.
The only thing you can do is check with your institution to allow you two simultaneous logins or to give you two separate accounts. Where I work we can log in from as many devices as we want with BlueSocket so I can be connected with my iPhone and my Laptop at the same time. So it must be a configuration they can do.