When I see "1 new answer" I always click it. Much of the time it fails; this is often, but I don't think always, before/after posting an answer of my own.
I've seen this happen at least once since gaining 2k on a beta site. In each of these cases, after reloading, no deleted answer is seen and a new answer is visible with the expected timestamp.
Later seen on MSO after following a comment notification on an answer already left; the page had been loaded about 51 minutes before the new-answer notification appeared, per a userscript's tracking.
Not seen (that is, the new answer loaded as expected) on RPG while drafting a new answer (which was not, as it happens, ever posted).
Seen again on SO while editing an already-posted answer.
Seen on ELL after finishing an edit to a question (without posting any answer at all).
Seen on ELL after loading a page and leaving it for half an hour.
Seen on M.SO after loading a page and leaving it for four minutes.
Seen on ELL after loading a page and leaving it for 53 minutes. On the same page at the same time, live refresh of an edited post did work as expected.
Seen here after following a comment and leaving other comments.
Why is this?
Mozilla/5.0 (Windows NT 6.3; rv:25.6) Gecko/20150723 Firefox/31.9 PaleMoon/25.6.0 — a Firefox fork on Win 8.1 x86. No other live-refresh functionality fails*, including new questions on the homepage using a very similar mechanism.
*Barring any idea that there are new comments in certain review queues, anyway.