Redmine

Well, you can tie files to a version of your project (though you will obviously need a version for that) , but that's all I know of.

I think we have too many different kinds of files for that to work, but thanks anyway! :)

Whoops, just found this--maybe an FTP plugin will do the trick: http://www.redmine.org/wiki/1/PluginEzftpsearch

The Plugin only allows you to search in FTP shares. Anyway, if you have some sort of shared folder (be it over ftp, webdav or whatnot) or at least a folder every person that needs to be able to write to has access to, you could use the filesystem repository scheme. You can show (though not upload through redmine) any folder on the server with that repository scheme. If you have that much files to manage, you might be better off using svn and e.g. tortoiseSVN on windows anyway.

True, but I really hoping that we would be able to have the file system directly integrated into Redmine so that you could just upload stuff into the appropriate folder using the "Files" tab.
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My programmer/server literate teammates mentioned the repository as a possibility, but thought that it wasn't as well suited for the types of files we're dealing with as an FTP server would be.
That isn't really the ideal solution we were hoping for though...It's always been a struggle for us to get people to upload stuff to the server, and I'll admit I'm not exempt from that. Who wants to have another password, download, install, and learn how to use an independent desktop FTP program, and then have to launch that and use it to upload files when it's 10X easier, simpler and more convenient to just attach a pic to a forum post or load it into a Photobucket account? The problem with that is that someday the team member disappears, their PC crashes, or the forum breaks down and all those non-uploaded, non-backed up files vanish.
So I think that what we're going to have to do is to just urge people to load their files into the Files tab, and then once a month somebody will move them all onto the fileserver en masse. Hopefully this will be convenient enough that everybody will start to use it. :)

And thanks again for your advice; the repository system was not something I would have thought of on my own. :) I thought it was only meant for code.

Well, one last idea if you have someone who knows his way around apache/dav: you can use the filesystem repository and make the same folder available through WebDAV, the user/password for the WebDAV can be the same as the redmine ones if you use Redmine.pm for authentication and authorization, and you can mount the WebDAV directories as network folders in windows/linux/OS X.

So you would get: a folder that is shown in redmine in the Repository tab and which you can read and write to through WebDAV/a network drive on your PC.

Hm, I'll have them take a look at this thread. Thanks for the suggestion. :) Anything that makes things easier is worth a try.