This is not the case. Given the overhead involved in processing a code-generated web page and serving image assets like avatars, websites today cache and reuse what they can. Your web browser rarely makes the servers go back to the actual database to get an avatar. The purpose of the dateline is to ensure that when a new avatar is uploaded distributed content services and web browser caches don't keep using the old one, because the URL has changed.
Too many secrets! If someone quoted your avatar (presumably in the ritual to vote for your post to be your tag) and it showed your old avatar for a little while than it's poor caching behavior or bandwidth saving or something? (this was in chrome)