Ok, in most cases, you can open Premiere Pro Projects going backwards with no problem. There are however "bugs" where things might not import in correct; specifically with transitions, and effects.
Premiere Pro CC Projects- will NOT open in any versions prior to CC. Period. They designed it this way on purpose to keep people out of the "Owning Software" and move them into the "Pay to Use" program.
However, I can from experience, that Premiere Pro CS5.5 will most likely open a CS6 project.
Lastly; having been a PPro user for many years, one project we had from a client required us to "jump through hoops". We had to take their project; which was (just an example) Premiere Pro CS4. CS6 would not open it.
However, CS5 would. So we opened it in CS5. Saved, then moved it into CS6.
If you are having problems opening a project saved in a later version in an older version- like you said you had at school: You can try:
1.) Try opening it in an even "older" version. Yes I know this seems crazy but sometimes it works. Then save, then move up to the version they have.
2.) You could try to export an EDL and bring that in to premiere; although the results could be not ideal.
3.) Try to use the "Reduce Project" feature where it clips all of you files, with handles, and let it render out and wrap the project into a single Folder. This will create new clips entirely, so make sure you select a good codec to output to avoid loss of quality by the rendering which will take place.
4.) If all else fails. And you absolutely must work on the project on your schools computer: Render out each video track as a video file, and do the same for audio. Render each track from 0:00:00:00. Then bring in each one and lay them on tracks accordingly.
The last option would require work, because transitions, opacity, etc, may get lost depending on your project.
Hope this helps.
SideNote: I would highly recommend to anyone NOT using CC. CS6 Suite offers nearly all of the same options as CC. I am still using CS6 (and my studio owns approximately 50 seats which is a fortune if we were to upgrade). There is really no reason to use CC over CS6 that I can think of, unless you need one of the very few upgrades it brought with it. We have 50 seats with CS6; and its paid for and done. With CC, we'd be looking at $3500 a month perpetually to use the suite. Obscene.