I was going to upgrade my site to 2.0.13, but I'm having problem upgrading CBB to version 2.2 (2.0.13 comes with 1.1, but I was using 2.2 with all kinds of extra features).
Fatal error: Call to a member function on a non-object in /home/jonr/public_html/modules/newbb/admin/index.php on line 165

I get this kind of errors all the time when I try to access any part of the CBB module. Should I downgrade/upgrade to some different version of CBB?

Hmmm....BACKUP FIRST! None of these guaranteed but if you are considering upgrading do a backup then

You could try
1. upgrade to 2.0.13 and see if that fixes it (or did you mean the problem came when you did upgrade to 2.0.13?)
2. Upgrade to latest CBB and see if that fixes it? (not sure but I think there is another version there)
OR...
3. Upgrade to 2.0.14 and go the whole hog for that latest release.

But remember to backup first!