If the Everett's many-worlds interpretation is correct, this could be happening all the time. But we can only experience one of the world lines where our universe does not collapse.
It's not an especially cheerful idea, to be honest. Even if that argument works -- and to be fair, this is about the best possible scenario for it -- it's too easy to generalize.
There's no particular reason you should limit your self-identification to just the other branches you exist in in this universe. The full argument is a bit oversized for this site, but.. consider that there are branches where you've got the exact same past as here (as far as you can tell), except that you're actually living in a simulation being run by some alien.
If you accept the notion of quantum suicide, and also provisionally this expanded version, then consider that even if the proportion of timelines in which you live in simulations (and thus, the credence you should give to such claims) is low now, a sufficient number of vacuum collapse events could actually change that over time.
When does it shift you to another parallel universe? At the moment your consciousness would get lost? At the moment your body gets wounded but your mind still experiences it? At the moment a bullet starts flying towards you?
What about dying from getting old, if "it" has to keep your mind alive, does it mean everyone eventually ends up experiencing a world where they get 10000+ years old while the rest doesn't?
According to this interpretation there is no "shift", just copies. In fact, this interpretation is attractive exactly because it doesn't have the weird feature of the copenhagen interpretation where the wavefunction collapses when it is "observed", whatever that means.
So a more correct interpretation would be : there's infinitely many copies of you, each experiencing slightly different event, and while most survive (an infinite number), every moment in time there's a few (still an infinite number though) die horribly every moment in time.
Sooner or later it will be one of you reading this. :)
O_O ...