I've seen it happen more times on SuperUser than here, but several times the page has barely finished rendering and then the notice pops up:
Maybe people like this instant notice that the site is "alive", but I find it annoying that I've just loaded the page and the notice is telling me that the information is slightly stale and I must click on the page to get a current version.
I think a slight delay would alleviate this "omg, my new page isn't new anymore!" experience. Maybe 10-15 seconds after page refresh before popping up the new activity banner?
If not a delay, maybe it would be smoother to just insert the new activity into the questions list if it happens in the first few seconds?
Based on Geoff's answer it seems the goal is to get these updates as fast as possible - but I guess I just don't see the purpose in telling people that their freshly loaded page isn't fresh.