How large is your document? Are you the only one using the credentials of the user that has this Named CAL assigned at all times? Can you post a screenshot of the Task Manager -> Performance tab when you run it on your server?
There you have it: every CAL can be used to maintain a single session only at any time. You may get the feeling that you can open a document 10 times using a single CAL, but in reality each new session will terminate the previous one (without telling you anything). As soon as you try to use a session that was opened before, you will get the "connection lost" message and the client will try to restore that session. And it will prompty disconnect the last session that was used immediately before.
You'll need to assign more CALs to more accounts. Usually one CAL to one separate account for each display. There is no CAL-sharing in QlikView.