On the Bug Reports page, Manfred noted that he was having problems uploading an avatar for his personal profile or for his group. I tried to upload an image myself and ran into some very strange problems: first, the image wouldn’t crop at all, and then it started bouncing around the page like the cards at the end of a just-won game of computer solitaire!
After browsing the BuddyPress forums, I found a post that described similar issues. One person mentioned plugin conflicts, which made me wonder whether we had any plugins that might be screwing things up.
It turned out that the culprit was the Litebox plugin for WordPress. Deactivating that plugin seemed to solve the issue.
So: you should be able to upload avatars without problems now! Please let me know whether you continue to experience issues!