Yes. Even if you get a new computer, you can continue to use your ADSM server name and your old node name with the network backup service on the new computer. Since you're using the same account, node name, you will have access to all your old files.
There is one quirk, though. If you are restoring files to a different computer or if you are restoring to a hard drive with a different label, then you can not select "Restore to Original location". Instead, you must select "Restore to Following location" and specify where to restore the files to.
Warning: Do NOT try to restore system files of old machine to the new machine. All PCs/Macs have their own system settings.
If your old computer is handed down to someone else, make sure that its new owner isn't using your ADSM setup. The new owner must register the machine themselves and get their own node name.
June 29, 2015