The wifi icon on the screen.
And in manage connections you have not had a green checkmark, right?
I don't know the reason. But if you've had the mobile network and wifi on when the other apps worked and browser did not as described in your original post, the reason is that those other apps were using the mobile network and not wifi. Thus when you turned the mobile network off, those apps could not connect.
For some reason you are not connecting to the RIM servers over wifi.
One other thought is to resend service books and see what happens. See
How do I resend my service books? - BlackBerryFAQ. Note you can do it from the handset in the email setup app.
Posted via BlackBerryForums.com Mobile