As far as I know, yes, but they will interfere with one another. Microsoft builds their systems so that they aren't backward-compatible. They usually do provide drivers for earlier versions, so you may be able to save in 2003 format in the 2K version.
You can save the database as an Access 2000 database, and you'll be able to work on it in Access 2003 without having to convert back and forth. One word of caution is that you have to make sure that your VBA code is valid for Access 2000, which is probably true 95% of the time if you don't real advanced programming.
Oh, I just saw that you're having problems with 2003. Try converting the 2000 db to 2003 and then back to 2000. I think Access made me convert older dbs to 2003, but when I converted backwards, they are still usable in Access 2003.
Just install them in diffrent directories. The default one is
C:\Program Files\Microsoft Office
Son install 2003 in this Subdirectory, then install 2002 in
C:\Program Files\Microsoft Office\2K