What's happening is that some application (quite possibly one of those asset management programs) is creating a file or folder on a disk volume, without checking first whether that volume is actually mounted. Normally, you would expect such a check to return an error message, but it's possible to specify "create any missing folders" as part of the request.
Now, suppose you have a volume named "Assets" and a folder on that named "Art". When the volume is mounted, the path to that folder would be "/Volumes/Assets/Art", and the folder might contain a file "/Volumes/Assets/Art/Mountain.jpg". An asset management program might remember either of those paths.
Then, at some later date, when the volume is not mounted, it tries to follow one of those paths. It could be looking to see if any new assets appear in the "Art" folder, or be trying to add a new asset to it, or be looking for a specific asset ("Mountain.jpg" in the example) that it remembers being there. Any of those should produce a "directory not found" error, because "/Volumes/Assets" is no longer present. If the program tries to suppress that error, by specifying "create any missing folders", it will create first the folder "/Volumes/Assets", and then inside that the folder "/Volumes/Assets/Art".
The programmer might have been thinking that the missing folder would be "Art", not realizing that a missing "/Volumes/Assets" would also trigger the error condition. It's a distressingly common programmer's blind spot.
But "/Volumes" is supposed to be reserved for mounted disk volumes. Ordinary folders should not be placed there. I'll refer to ordinary folders inside "/Volumes" as "phantom folders". When you later mount the "Assets" disk volume, the OS notices that "/Volumes/Assets" is already there, and mounts it instead at "/Volumes/Assets 1".
Your asset management program eventually notices the folder "/Volumes/Assets 1/Art", and remembers that path. The next time it looks while the volume is not mounted, it creates a folder there. When you mount the volume next time, "/Volumes/Assets" and "/Volumes/Assets 1" are both taken, so it mounts as "/Volumes/Assets 2". Repeat weekly to get your observed behavior.
If all the program is doing is looking for new assets, the folders it creates will be empty, except possible for nested empty folders. But if the program is trying to save assets, it's saving them to these newly created folders, where they're just getting lost.
The first thing you need to do is find out what's on the phantom folders in /Volumes. If there are any assets in there, you will probably want to copy them to where they belong. Seeing what's being created in the phantom folders might give you a clue about which program is doing this. (Normal Snow Leopard permissions on "/Volumes" will force all the phantom folders to be empty, though, so there may not be any clues to be garnered.)
/Volumes is normally hidden, but you can examine it in Finder by invoking the "Go -> Go to Folder..." menu command, and typing in "/Volumes" as the name of the folder to go to.
All of your mounted disk volumes should appear there. Finder will identify them as aliases, even putting the curved-arrow alias badge on their icons. Get Info will identify them as aliases. The phantom folders will be lacking the alias badge, and Get Info will identify them as folders. You can browse inside them using normal Finder techniques.
After you've reclaimed any misplaced assets in those phantom folders, delete them. (You may need to authenticate with an admin password to do that, although the user that created them should be able to delete them.) The next time you mount the disk volume, it should mount in its correct location.
Try to identify the misbehaving program. I suspect one of your asset management tools, but it could be something else. Another possible candidate is a program that is trying to remember recently used documents, and is remembering a little too aggressively. Stop using it, or at least don't use it while the disk volume is not mounted, and file a bug report with its author.