Is it possible that you have opened older files created in CS5 or earlier? That in itself is not a problem, but if the file wants to keep that earlier version compatability when you save, then there's a mismatch between CS6 and earlier versions of Illustrator, especially around transparency issues like drop shadows.
Make sure to save as an illustrator .ai format and on the second screen of the save dialog box choose CS6 as the version.
Not sure if that's it, but worth checking. Hope that helps…
Adobe Certified Instructor
Unfortunately all the files are native to CS6, and saved as version CS6. Thanks for your thoughts, though!
...native to CS6, and saved as version CS6.
.AI or .EPS?
Are you fully updated? I had a similar problem when I first loaded CS6. The problem disappeared after updating.
If that's not it, more info on operating system and version, hardware, saving method (local or network), etc. might help someone diagnose the problem.