I've seen these too, but not as common. The issue is that these can also appear if BF3 crashed.
Try to rule out BF3 crashing (we can't even tell when it crashes or not, a disconnect and crash look the same). Close all programs, run a chkdsk and all that. Then when you go to play join a server and don't touch anything. Let it connect and open, see what happens.
If you don't think it's a game crash, try connecting directly to your modem (I assume you're connected to a router) and see if that helps. If it does, the router sounds like the issue.
Best Answer
That's the in game chat system which you can open by pressing the Enter key.