Dell M6600: "No internet access", can browse internet, can remote into it, cannot activate office 2013, cannot join domain
Dealing specifically with the "No Internet Access" message, Windows uses NCSI (Network Connectivity Status Indicator) for this. This checks two things:
A request for http://www.msftncsi.com/ncsi.txt - Page called ncsi.txt containing the following line of text with no terminating new line or other non-printing characters: Microsoft NCSI
A request for DNS name resolution of dns.msftncsi.com - Resolution of the DNS name to: 131.107.255.255
If either of these fail, you get the "No Internet Access" message. I would guess that Office isn't even trying to register at this stage, you're getting a stock message based on NCSI having failed. I'd fix the NCSI issue first then try and register Office again.
We had a similar issue with a couple of users with admin rights on their machines (developers) who were setting up their machines, including installing Office 2013. One connected to his home network and activated Office that way, while the other contacted me.
Our small office uses a Watchguard router that includes a proxy server that requires no settings on the user end. I added "allow range requests through unmodified," to no avail. I then told Watchguard not to proxy that user at all. But because his install had previously failed, I had to wander around his hard drive and registry deleting stuff before I could get it to work.
I know you said in reply to one of Massimo's answers that there's no proxy for the network, but it might be worth double-checking if you're not 100% certain.
The issue turned out to be driver related. (DOH! I forgot to try a different driver version when uninstalling and re-installing the drivers.)
The OEM drivers caused the machine to have the "no internet access" symptom, and using a driver straight from Intel resulted in the system functioning correctly. The system is a Dell M6600 with an Intel 6200 WiFi card. The issue returned after I installed the WiFi driver from Dell, and I had to do a system restore to get the issue to revert, then install the driver from Intel, and the WiFi worked, as well as everything else.
Thank you everyone for the excellent suggestions!