I have upgraded to 16.04 LTS (from 15.10). Wi-Fi and LAN both shows connected, but internet is not working.
I have done re-installation of Broadcom wi-fi drivers (through additional drivers), restarted network-manager services. But problem not solved.
Sounds like a DNS problem...
Check if you are able to connect to your router IP. (if you can, it probably is)
Check your ifconfig for DNS and Gateway info.
If it is a DNS problem, recheck your network configuration on DHCP and/or use a static lan address with your router/dhcp server as 1st DNS, and other DNS servers following.