What could have changed to make this happen?
Have you "updated" your QuickTime since the last time you played the game?
You might be able to fix the problem by completely uninstalling all versions of QuickTime, running a registry cleaner (like the registry "Scan for Issues" of Ccleaner) to remove all traces of the previous QuickTime install, and then reinstalling the version of QuickTime that shipped with the game.
If you don't want to uninstall QuickTime from your main computer, RHEM 4 will run in a virtual machine like VMware, VirtualBox, or Virtual PC. So if you've already set up a virtual machine for older games, you can just install and play in that.