Wow. I literally have this exact same problem, and it's driving me crazy trying to resolve it!
I have Fusion 5.0.3, and have been using it for quite awhile, just fine with OS X Lion using the NAT adapter. I've been running Windows 8 in the VM too.
Now, I've upgraded to OS X Mavericks (and also Windows 8.1) and the NAT adapter no longer works. However, the Bridged setting works as expected, getting a different IP and all that. And, just like Dave, the NAT does work just fine at my office, but not at home (where I also need it for VPN). And also like Dave, I have AT&T U-Verse with their standard router/modem combo device.
I have tried both wired (ethernet) and wireless, and neither gets this working. Obviously I know the configuration works, because it works at my office. And of course, it's been working just fine for over a year now before I upgraded. And the strangest thing, is that after playing a little shuffle game with disabling settings and adapters and re-connecting things, it will occasionally (but not often) actually work for a brief second (long enough for a single web page to load), but will then quickly go back to not having internet connectivity.
It would certainly seem like some sort of network-related issue, since it does work elsewhere; although I don't what would have changed in OS X to cause my home network to no longer work with Fusion. Does VMWare (or anyone) have any idea what might cause this, and maybe even a way to fix it? Or is it confirmed that upgrading to Fusion 6 will solve the problem? I'd rather not spend the money to upgrade, but if I can be guaranteed that things will work like they're supposed to, it might be worth it.
~ Sean