I know JA templates work fine in Joomla 1.5, as we have many users who use them. However, we haven't done any testing with Joomla 1.6 with a JA template, so not sure what it might be. However, I'd recommend looking into the following possibilities:
1) I definitely think you should consider upgrading to Joomla 1.6.3. 1.6.0-1.6.2 had quite a few security problems, to begin with, but there were also many, many bugs (in the area of 200) that were fixed in those 2 releases. It's very possible that this issue was fixed in a newer Joomla release. We only tested JFBConnect with 1.6.3, and only recommend using that revision.
2) If you still have the issue, try re-ordering the JFBConnect System plugin to the first or last positions. It's possible that, if there's a JA system plugin, if it executes before (or after) ours, it will clear up the issue.
Hope that gets you started, but if you continue to have issues, just let us know!
Alex