FDR's scheme to pack the Supreme Court was too clever by half. If he had made a straightforward case that he wanted a couple of additional seats on the Court to stop legal obstruction of the New Deal, he might very well have gotten away with it. It is not unprecedented in American history for the number of SCOTUS justices to be changed for overtly political reasons.
Unfortunately (or fortunately, depending on your perspective) FDR was too sneaky, and too greedy, for his own good. He claimed that the Court needed to be expanded because the incumbent justices were too old, and unable to keep pace with their heavy workload. This meant that when Chief Justice Hughes demonstrated the Court had no problems with its current workload, the public case for adding more seats collapsed. The ruse was too transparent, and made the public feel like FDR was trying to hoodwink them. Adding six extra justices to a Court of nine also struck people as excessive, even if they might have been open to a smaller expansion.
Even after Congress initially rebuffed him, FDR might still have gotten legislation adding just two more seats to the Court. He had a strong ally in Senate Majority Leader Joe Robinson, who had been promised one of those seats for himself. But when Robinson died of a heart attack, the fight was over.
FDR was too cunning to leave the matter there however. Shortly after the Court packing fight, he slipped legislation through Congress which significantly increased the pension for retired judges. This duly produced a rash of retirements from the Supreme Court. By the time FDR was sworn in to his third term, seven of the nine justices on SCOTUS were his appointees.
One of the interesting minor subplots of the Court packing affair was the boost it gave to the career of a young Texas politician named Lyndon Johnson. He ran for Congress on a platform strongly supporting Roosevelt in the fight with the Court, and the issue got him elected. FDR noticed and appreciated the support, and took the young LBJ under his wing as a protégé. With great consequences for subsequent American history.