I guess it depends on how much you weight the theme of "law and order just collapsed" vs. "our predecessors were geniuses and we keep finding their stuff."
Just like with defining science fiction, I won't fight too hard to gatekeep what is and isn't post-apocalyptic as I think the lines can be pretty fuzzy. Let's take a look at a few properties and see where they stand up.
In
Mad Max, Max Rockatansky was a police officer serving during a time of great civil unrest, but the state still existed and evidence by the existance of the Main Force Patrol who was charged with maintaining law & order. They even provided Max with a sweet ride in the form of his Interceptor (last of the V-8s). The first movie is apocalyptic but not post-apocalyptic. But 1981's
The Road Warrior is different
, we see the aftermath of the total collapse of the state with the only sign of civilization being the settlers at the oil refinery. Definitely post-apocalyptic.
The Walking Dead (television series): This one seems easy enough. Post-apocalyptic, right? For the most part, yeah. The state has collapsed in its entirety with the remaining vestiges of civilization being relegated to pockets of settlers scattered here and there. But towards the end of the series, our pit stained heroes come across the Commonwealth. The Commonwealth has a government that provides civil services, a codified set of laws, lawyers, a military, a cash economy, they produce fuel and ammunition, a population of 50,000+, and they're stable (not even our heroes could destroy it like they did Alexandria). I would argue at this point they moved away from the post apocalyptic grind.
Rifts: In Palladium's RPG, the game is set more than 250 years after the titular rifts devestated the planet. You've got several state actors including the Coalition, the Federation of Magic, Lone Star, Free Quebec, etc., etc. This isn't really a post apocalyptic game though a lot of people think it is.
Dungeons and Dragons: Given the number of dungeons you delve that were part of ancient civilizations, I don't think it's unfair to put D&D on the list. But even in Dark Sun, the state exists. There are wild frontiers filled with monsters, remnants of ancient civilizations, and treature, but civilization is almost always close by. Not really post apocalyptic.
I think a good rule of thumb is its still post-apocalyptic until its climbed back up to the equivalent of the prior civilization. As far as I know at no point in the FO timeline are you back to global communication and transport, so...
That's not a bad rule of thumb. I think once the apocalypse is no longer within living memory and you have a defacto state then it's no longer post-apocalyptic.