The Jetsons/WMG

The series takes place in a post-apocalyptic world in which the surface has been ravaged.
Think about it for a moment: they have flying cars and sentient robots, but their computers are way outdated, their culture is more primitive, and the Internet is missing.

The truth is that the world the Jetsons live in is a a world where the planet was doomed in some way. Now they, and all the other people there, live in these buildings in the sky to protect themselves from the mutants. They don't know how their technology works; they live in a Scavenger World and just try to make the technology of their ancestors run as best they can. The AIs that they use are mixed with their own primitive attempts at computers, creating the odd robots and computers that exist in the series.

Further proof: In one episode, a criminal says he'll be happier going to prison, which is located on the surface.
 * They do show a brief glimpse of The Surface in the movie, and it is a polluted mess.
 * about the computers/Internet thing: you DO realize this series was created in the early 1960's, right?
 * It's the future of Fallout.
 * No, it's the future of Tengen Toppa Gurren Lagann.
 * No, The Jetsons is right before Fallout
 * You're all wrong. Global warming has flooded the surface.
 * Again, we've seen glimpses of the surface. Even the parts that aren't underwater look gloomy.
 * Source for the glimpses of the surface?
 * A glimpse of the surface is shown in the episode "The Flying Suit"

Addition to the above theory: the apocalyptic event was a botched Cuban Missile Crisis.
It would fit with the context of when the show first aired for it to be World War III: this was during the Cold War, 1962-3. When did the Cuban Missile Crisis occur: 1963. The Jetsons may be what might have happened if the Cuban Missile Crisis turned the Cold War hot. The reason we've only seen white-bred Americans is because their ancestors were the only ones in a wealthy-enough class to escape. The sky cities are a secret project by the government to deal with the worst-cased scenario.

The Flintstones are on the surface.
How did it happen? Who knows. But we might want to start checking the background of old Flintstones cartoons for big metal poles.
 * Jossed in the Made for TV Movie The Jetsons Meet The Flintstones. The Flintstones canonically exist in the Jetsons' past.
 * Or alternatively, the people in the Flintstones are actually descendants of people from the Jetsons' future who escaped their decaying world by travelling back in time. All the talking dinosaurs were created through bioengineering to serve them.
 * This could also explain why the Flintstones' technology is basically a "stone age" version of modern technology (okay, 1960s), rather than actual Stone Age tech. There's been some kind of apocalypse after the events shown in The Jetsons (perhaps a meteor strike or nuclear war), and they're just trying to rebuild up to the level they once knew.
 * Or alternatively, Elroy screwed up, and what he thought was a time machine is actually a transporter which got them to the planet's surface.
 * Alternatively, they take place back in the past, but not in the Stone Age. See the above: The Flinstones could be set on the Earth before it became nigh-uninhabitable.
 * If this is the case (and Elroy's time machine really did just send them to the ground), then maybe their planet is really a Feral World in Warhammer 40000? The Jetsons' magic sky castles can be the home of the nobility who govern the planet, while the primitive surface communities are Space Marine recruitment areas. Think about it, if Bam Bam isn't going to be prime Astartes material when he grows up, then i don't know who is. This also adds the possibility that Gazoo is a xeno hiding from the Imperium and decided he might as well screw with the primitives.
 * The Flitstones are future Amish!

Cogswell is a descendant of Mr. Slate.
They look exactly alike. In an episode of The Flintstones where Gazoo sends the gang to the Jetsons-like future, they meet a descendant of Fred's boss Mr. Slate who almost dresses like Cogswell.
 * This becomes especially obvious if you watch The Jetsons Meet the Flintstones, which features both characters with the similarity between the two never noted.
 * Or Cogswell IS Mr. Slate.

George and Jane had a Shotgun Wedding.
For anyone who can do math, George and Jane may or may not have had Judy when they were teenagers. In this case, it would be logical to assume that they were forced to drop out of high school and get married. It would explain a lot. George is stuck in a worthless job because he can't get any other job due to not having a high school diploma. Jane never learned to drive because being pregnant at sixteen didn't give her a chance to gets her license. And Judy spends most of her screen time lusting after boys -- clearly she inherited her parents' sexual genes.
 * Unlikely, either way. While H-B didn't go for a family aesthetic exactly (not at first, anyway), they weren't exactly adult oriented either, prime-time airing or not. For this reason, the above would clash with who they were as animation producers, and what kind of shows they usually turned out. The whole age difference between Jane and Judy was likely an accidental oversight on H-B's part.

The show takes place in an Alternate History where the Civil Rights movement led to the mass extermination of non-white people.
Notice how there are absolutely no non-white characters in a supposedly futuristic setting (albeit one which currently would be in the past. Also, notice how the only vaguely black character is both a robot and a domestic servant. One could infer that the U.S. government of this timeline responded to the Civil Rights movement by creating huge, skyrise cities and flying cars, transporting all "acceptable" people into them, nuking the surface, and living on in their advanced utopian society served by robots based on their ideal version of other ethnicities. This troper actually came across a dark-skinned character in one episode involving awards so this is less likely.
 * How in the hell is Rosie "vaguely black?" She doesn't look, act or talk like any black person who's ever been seen, heard or told of by third parties anywhere on this planet.
 * She's partly based on the maid from Fibber McGee and Molly,.
 * I think you're referring to Birdie on The Great Gildersleeve, a spin-off of Fibber McGee and Molly. In any case, The Other Wiki says she's based of Hazel from the series of the same name.
 * Who says they need mass-extermination? You get the same results just by leaving everyone on the ground alive.
 * This troper always thought Rosie was a stereotypical jewbot. Maybe it takes place in a world where the Nazis won WWII and jewish people were turned into robots. and those robots were turned into slaves.
 * And the jewbots became THE CYBERMEN

Judy and Elroy are adopted.
Both parents are redheads, and they have children with blond and white hair?
 * Perhaps Judy or Jane has experimented with hair dye. (It doesn't seem like something a man or young boy would be into.)

In The Movie, Judy was going through a phase where it was cool to make your voice sound like a popular singer of the era.
This should explain Judy's voice sounding like Tiffany instead of Janet Waldo.
 * I guess she wanted to impress Cosmic Cosmo.
 * This troper didn't noticed the difference.

The family and other characters are related to The Flintstones.
Think about it.
 * George=Fred
 * Jane=Wilma
 * Elroy=Barney
 * Judy=Betty
 * Astro=Dino
 * Mr. Spacely=Mr. Slate