That is strange. It shouldn't be possible as the NAS is not involved in traffic between any client and an outside server. So either the NAS is generating so much 'noise' on the network that it causes regular traffic to drop, or somehow the NAS changes the network topology.
Did you install any additional software on the NAS? What is the IP address of the NAS? Are it always the same websites which work and which don't work?
If you are a die-hard, you can install Wireshark
on you computer to see if there is spurious network traffic originating from the NAS, and to see what exactly is the difference between a working and a not-working site.