Please try re-installing the correct driver:
sudo apt-get install --reinstall bcmwl-kernel-source
If there are errors, please edit your question to report them.
First, check the settings in the router. WPA2-AES is preferred; not any WPA and WPA2 mixed mode and certainly not TKIP. Second, if your router is capable of N speeds, you may have better connectivity with a channel width of 20 MHz in the 2.4 GHz band instead of automatic 20/40 MHz, although it is likely to affect N speeds. I also have better luck with a fixed channel, either 1, 6 or 11, rather than automatic channel selection. Also, be certain the router is not set to use N speeds only; auto B, G and N is preferred. After making these changes, reboot the router.
Next, I recommend that your regulatory domain be set explicitly. Check yours:
sudo iw reg get
If you get 00, that is a one-size-maybe-fits-all setting. Find yours here: http://en.wikipedia.org/wiki/ISO_3166-1_alpha-2 Then set it temporarily:
sudo iw reg set IS
Of course, substitute your country code if not Iceland. Set it permanently:
gksudo gedit /etc/default/crda
Use nano or kate or leafpad if you don't have the text editor gedit.
Change the last line to read:
Proofread carefully, save and close the text editor.
Next, I'd set IPv6 to Ignore in Network Manager: http://docs.fedoraproject.org/en-US/Fedora/18/html/Installation_Guide/images/netconfig/network-connections-ipv6-ignore.png This example is for ethernet, but you want wireless.
Is there any improvement?