No, not like what you describe. People from different states may make fun of each other for political or cultural reasons (blue-state libtards, red-state hicks, etc.), and there may be disputes between state governments over water rights or other issues that rarely make the news. There have been some armed standoffs between locals and what they regard as overreach by the federal government, as in the Bundy Ranch case some years ago. That clearly was a piece of great stupidity on the federal government's part.
Of course, 156 years ago, there was the American Civil War, which was a war between states, and was by far the worst war in American history, in terms of deaths (over 600,000) both in absolute numbers, and as a percentage of the population, which was then only about 30 million.