Westeros was kind of due for a major breakup either way. It's based loosely on the United Kingdom except maybe 3 times bigger and with 8 major factions, the number perhaps moving up or down over time. It's a continent with thousands of years of history, and only in the last 300 years has it been united under a single rule when the Targaryan's conquered and united it with dragons making the 7 kindoms/8 great houses kneel to them.
While Robert's rebellion had justification it's entirely possible that you had more going on at the time and the Mad King had a right to be paranoid. While his family had 3 centuries of rule they dragons they initially conquered with are long gone, so it's entirely possible that cracks were forming and there was a conspiracy to topple the Targaryan's. We don't know how often great families intermingled, but before the rebellion we had a Stark and a Baratheon squiring together under an Aryn and becoming bff's plus a number of marriages being arranged: Baratheon-Stark, Stark-Tully, Aryn-Tully. That's a 4 great house alliance whose positioning conveniently sandwiches Kingslanding in a pincer attack, with the Baratheon's coincidently having the best claim to the Iron Throne in a post-Targaryan Westeros. Then when the Mad King through a torch on the dry tinder kicking off the rebellion, the Lannisters through there hat in with the Baratheon's and you had an alliance that couldn't be denied by the remaining great houses.
That's old history though which just left another ticking time bomb - the alliance was strong through the rebellion but how do you keep such a multifaceted object together through the generations? In GoT we see the Baratheon & Lannisters split and the Aryn's withdraw, leaving the Stark's & Tully's alone and the Tyrel's able to pick their side and play Kingmaker. I mean this is all fictional of course, but I've always felt the way it was setup without "Fire and Blood" a fracturing of the Kingdoms back into their original state was inevitable.
Ultimately it all comes down to power, but it's just to difficult for any non-Targaryan to build and maintain an alliance strong enough to keep everyone else in check. Just look at the Greyjoy's as the canary in the coal mine. Immediately following Robert's rebellion they split and declared themselves independent, Robert sent his fleet and forcibly brought them back into the fold. When Robert died and Joffrey succeeded, they immediately rebelled again... except through the duration of the show there's been too much infighting so they've been able to run amok as they please.
