Backups to external hard drive fails for some machines when replaced

[MockY]

I have an external hard drive installed on my file server with a shared folder where all the backups goes to every day. The backups are mainly from the file server itself, but also holds backups from various computers around the office. Since the backups are totaling a size that is not manageable for remote upload, I simply grab the hard drive every Friday and take it home, and subsequently copy it over to a server at home so that the data is stored on a remote site as well.

This got pretty old pretty fast so I got myself an identical external hard drive (in reality, it's an identical enclosure with an identical hard drive in it) so I can simply swap it every Friday instead. Much more convenient.

The problem is, even though the setup is identical, as in the external hard drive has the same shared folder with the same permissions, some of the office computers simply can't write to the share anymore, which leads to failed backups.

The share still shows up for these machines but when trying to create anything in the share, they get the following error:

"Unable to create the folder <name>
The System cannot find the file specified."

Everything is identical, so why is this occurring and what do I do to prevent this from happening?

The system that houses the external hard drive is running Windows 2003 Server Standard Edition.