You can copy a project and paste it, but all the related files need to be in a single folder. Copy that folder, or all it's contents (files and sub-folders), and paste it somewhere else. The new files will keep their references within that main project folder, and shouldn't reference any of the files from the original folder. I don't highly recommend that method, but it will work.
As far as messing up references when moving single files, it's perfectly understandable that that can cause problems if not done properly.
Glenn that is what we have tried to do. All our files occur in the folder or subfolders yet when we copy and paste them elsewhere sometimes they keep the references.
I have even gone as far to copy and paste them onto my desktop from the server, disconnect from the server, open files, ensure references are good, save file, reconnect to server reopen files and magically it starts referencing files on the server from old location again.
I don't get why but it is very frustrating and keeps screwing up our projects if you aren't careful and make sure to fix the references.
Moving the files using SolidWorks Explorer will reduce the chances of problems occurring. It is free.