Well Matt I gave it a shot on my 2010 with no luck, and even on my old 2000. Still a no go. Are you sure this was an AutoCAD dwg? Apparently some other programs use the .dwg extension but they aren't AutoCAD.
A few years ago I upgraded computers at home and used one of those programs for transferring supposably everything from old to new, including the operating system. It didn't work and I had to install a new one from scratch. Of course a lot of stuff did transfer but under the old attempt. Now it thinks I have 2 systems, and won't open any file from the old system, even though they have the same file extensions. Is your situation similar?
I hope my info was useful, but if by some odd chance it actually solved your issue, feel free to mark your post as "Accept as Solution". Thanks! 🙂