Not remotely supported by history. Breakaway states routinely come to exist in times of trouble, particularly ones wtih preexisting history as an independent realm.
Robb Stark's aims were never to conquer southern Westeros, but to re-establish the Kingdom of the North which his ancestors once ruled, and annex to it the Riverlands (the territory of House Tully, which was supporting the Starks but unlike them have no traditional claim of kingship).
This is similar to the aim of Balon Greyjoy. The Iron Islands also were once an independent kingdom, and breaking away from Westerosi control to resume their murdering campaign was sort of his whole thing.
One can say that Robb Stark's position was short-sighted. It's absolutely not
what his father was going to do, but then, his father was aware that Stannis Baratheon was Robert Baratheon's rightful heir. As far as Robb knew Joffrey was Robert's rightful heir, but he could owe Joffrey no fealty, and the other brothers are also in the wrong so screw it KING IN THE NORTH.