Thank you JSanders! I'll look at the Opera Mobile, maybe I'll like it better.
In the mean time, the problem is solved
Here is what was wrong:
I have two routers (long story) - one is wired, a switch behind it, and behind the switch I have a wireless router.
Turns out, the DHCP in the main (wired) router was limited to 6 clients, thus when the BB connected to the network it acquired the IP from the wireless router which serves as a hotspot.
I changed the number of clients in the main router, disabled DHCP in the wireless, and now BB is connected with a green mark and browser works just fine.
Thanks again!