I was working on a project in school last week and took all my files home on my hard drive. The computer I used was a PC with Premiere Pro Version 6.0.3. My computer at home is a Mac. I signed up for Creative Cloud yesterday and was able to open the project with no problems. The only issue was the project I saved last week was on an older version of Premiere Pro and a newer copy was saved(I'm not sure what version of Premiere is on my Mac). I did some more work in this project and saved my work constantly(My Auto Save settings save 20 versions every five minutes). I come to school today and the project I was working on yesterday will not open and gives me the error "The project appears to be damaged, it cannot be opened."
Is it possible that this "older" version of Premiere cannot open files saved under the newer version?
Yes. An older version of Premiere will not open a project file from a newer version. I'm not used to the "project appears to be damaged" error in this case, but you're probably right on with the diagnosis.
You can move at least some elements of a project from a newer version to an older version by exporting Final Cut Pro XML from the newer version and importing the XML into a project from the older version. But effects and many other Premiere-specific settings won't translate into the XML. XML will work just fine as an interchange for rough cuts and media files, but you'll want to work in one version of Premiere once you start even the most basic finishing work.
Have you tried importing your project in from the Media Browser or at least its assets? In Premiere Pro CC you can also access all of your projects assets from the Media Browser. From the Media Browser, navigate to a project and select it. This will show you all of your contents within your project clips, graphics, sequences ect...
Hope this helps!