Are there any realistic means by which the southern states rebel against the national government in the time from 1900 to 1999? I thought of the Little Rock incident in 1957 might be plausible. I'm not asking whether the South would actually win, but if it is possible that they would try to recreate the CSA?
No. To begin with, the Upper South (NC, TN, VA), which was reluctant about secession in 1860, is about 40% Republican by 1900. Tennessee even elected a Republican governor in 1910, 1912, and 1920.
Second, the Federal government didn't do anything about black civil rights between 1900 and 1950, so white Southerners had no incentive to attempt secession or any other kind of rebellion then.
By 1950, Florida and Texas had become demographically diversified (especially Florida, with migration from other regions and immigration from abroad), and the new residents were not fanatic white supremacists.
To a lot of Southern states would never participate in any such movement.
The remaining Deep South states and Arkansas were still "Solid Southern", so to speak. But even there, the white supremacist tradition had been nibbled away. Lynching had practically disappeared by 1950, for instance.
But most importantly - the secession crisis of 1860 reached the critical level because a lot of white Southerners lived among slaves that outnumbered them, and were in terror of slave insurrection. Those Southerners believed that a Republican President was an "existential threat" requiring the drastic remedy of secession.
The overthrow of the Jim Crow regime in the 1950s and 1960s was painful, but not scary.
However - one possible scenario (very loose and tentative):
Zangara aims better and kills FDR before he can take office. Cactus Jack Garner hasn't a clue about checking the economic collapse, much less restoring confidence. Food riots break out in late 1933. Somehow this leads to radical takeovers in many areas and in Washington by 1936; rival revolutionists control other areas in uneasy cooperation. The U.S. is de facto balkanized. The People's Technocratic Republic announces a radical program which includes mandatory atheism and forcible racial integration. That's too much for the South, which breaks away.
(I'd like to find some mechanism for secession that would be based on immigration. In 1860, "the South" was the area where there was slavery.
In 1910, the South was the area where there were no immigrants.)