I will one up you, I don't put in notice until I'm 1-2 weeks into my new job. Don't really have any justification except for paranoia, which is only getting more validated from other anecdotes in this thread.
I would be onboard to try something like that, but how is this practically possible? you have to pose for 1-2 weeks like you're working both jobs, and then the 2 more weeks for the notice? how is that possible?
That seems just as risky as quitting the old job, if not moreso. One of the few things employers will confirm on a background check is employment status and period. If the new employer runs the check late and finds out you'll lose both jobs and burn two bridges.
I think they are pretty used to stale information due the batch nature of these things. That is, employment status showing "current" probably wouldn't raise any eyebrows.
But then you're not really giving notice. Presumably the purpose of giving notice is to give the company a couple weeks of grace period to transition your work to someone else while you're still around. (It depends on the situation of course. No one may care if you come back from a vacation and quit.)