I might be wrong and this might need further testing and confirmation from other people but it looks like when someone attempts to change his avatar but fails due to image size limits or any other problem then the avatar isn't changed but in the "Recent Activity" tab for that member you see "user x has changed his avatar".
If the member tries to change his avatar again and again but keeps failing then multiple instances of "user x has changed his avatar" will be logged, each one for each failed avatar change attempt.
I've just tried this locally in a variety ways and I can't even get one to log erroneously, let alone repeatedly.
Do you have any more specific details? What is the exact error the user is getting? What is the avatar_date for the user in the xf_user table? Do you see multiple entries in xf_news_feed for the user_id (content_id), content type "user" and action "avatar_change"?
What have you done to confirm and reproduce this problem on your board?