There's a problem between Eagle versions 5 and 6. In version 5, a binary file format was used. In version 6, it was updated to a XML format, so they are totally incompatible (all three files: sch, brd and lbr). So if you open a version 5 file in version 6, and save it, you'll not be able to open it again in version 5.
The same doesn't happen between versions 6 and 7. If you open a version 7 file in version 6, some error will appear, but you can view the file.
Also, you could install version 7 along version 6 only to see designs from other people, but you'll have to ensure version 6 as the default program for Eagle files so you do not mess with your version 6 files every time you open one.