Yes. They're ordinary computers running some variety of UNIX. If the machine is configured to allow this, then yes -- absolutely. The computers at my university had a similar arrangement which I often used for just that, though the university placed careful restrictions on who was allowed to login from where.

May I shift your perspective a little? Try imagining it, not as a "computer" in the Windows sense, but a machine that sits there and runs "jobs". When you run a program, that makes a new job, and you can have many jobs running at the same time belonging to many different users. It keeps careful track of which users are allowed to run which jobs and access which files but beyond that, the fact that you're the one sitting in the chair in front of it is largely incidental. You could remove the keyboard and monitor and just use it completely remotely -- I do for my UNIX machine, which is handy since it's a web server hundreds of kilometers away from me. Quite probably. UNIX does not have drive letters, as you may have noticed. Instead it has one(one per computer!) giant directory tree starting from one "root partition" -- that's going to be the hard drive of the computer you're sitting at -- and attaches or "mounts" more drives to different directories. Usually, the /etc/fstab file tells a UNIX computer which drives to attach where, when. Also try 'df' in a shell to show you what disks are mounted where at that particular moment.

I've seen something like your setup before. How it works is, each individual computer has it's own independent disk space used for system files(stored on it's root partition), so it can boot up independently. But the directories you use, something like /home/username/, get "mounted" to the big server through the network. When you login, do things, and logout, you're running programs on that particular machine, but all your data and settings gets saved to the server. So no matter which one you login to, you get the same files and settings.