The DVD install disc for Windows 7 (regardless of edition) cannot be used to upgrade an existing Windows 7 installation. Microsoft wants you to use the Anytime Upgrade feature to do in-place edition changes.
I'm assuming you came into owning a valid Ultimate copy (or at least a key for one). I've known students to get them through Academic Alliance and such and wanted to upgrade their current PCs which usually already had Home Premium on it.
Since it's a full retail product key, it doesn't work for Anytime Upgrade, and putting the disc in will probably only give you the option to clean install. This will backup the current Windows directory and install a whole new one. The problem, as you no-doubt realize, is that this means you lose all of your registry settings and user profiles. Your applications and games will still technically be on the computer, but Windows won't know about them since as far as it is concerned it's a fresh copy and you've not installed anything.
There is a trick, however, that can fool the install disc into thinking you already have a given edition and it'll let you do a repair install. This effectively upgrades you. Many users of the Windows 7 RC used this to "upgrade" to the retail version once released even though Microsoft officially said they'd need to start fresh with a clean install.
You can find a guide to the above trick here:
http://tech.icrontic.com/articles/upgrade-the-windows-7-rc-to-retail
The above trick only works for upgrading a Release Candidate install. If you have a retail or OEM Windows 7 (such as Home Premium) installed it will not work.
Also, all Windows 7 DVDs contain the full Ultimate edition and can (and do) install any lower edition. There is a single file on the disc that tells the installer which edition it should install which is why you don't get asked during setup. If removed, the installer will let you pick. This is useful if you buy or receive an Ultimate edition product key but not an Ultimate DVD and you don't want to have to download an entire copy (this can take quite some time on slower connections). If you still have your Home Premium DVD, you can still use the Ultimate key with that and install the Ultimate version. You'll have to use both the trick mentioned above as well as this trick:
http://windows7center.com/news/how-to-install-any-version-or-sku-of-windows-7/
It turns out that the upgrade to Ultimate seems to have actually worked, which is why Anytime Upgrade complains that my license isn't valid for further upgrades.
I don't know why the initial upgrade reported a failure, but anyway it all seems to be working fine now.
Best Answer
Yes, you can use Anytime Upgrade to upgrade a single machine.
When using Anytime Upgrade you receive a new product key that works in conjunction with the existing key.