Judicial veto points seem unlikely to bring down the United States. The judiciary has neither blocked needed political action, nor imposed great costs upon American business. Peter argued in comments that it is possible to imagine a world in which lawsuits block everything and make legislation impossible to implement, but that hasn’t happened anywhere else and really shows no sign of happening here.
What about America’s most famous veto point: the President? It is hard to envision a scenario in which this feature, by itself, could bring down the political system. Congress can override it, and the voters can toss the President out every four years. One can imagine a disasterous Herbert Hoover type vetoing legislation as the country sank into crisis, but it's hard. Most of his vetoes were innocuous. He vetoed the Tariff Act of 1930, a counter-productive piece of legislation if ever there was one. He also ixnayed the Emergency Relief and Construction Act of 1932, but a compromise measure sailed through. Congress overturned his other big vetoes. In short, Hoover wasn’t stopping an activist Congress from fighting the Depression. I suppose that one could imagine a situation in which a do-nothing President manages to let a crisis burn enough to have an even worse President elected in his or her stead. Harry Turtledove tried. But it is a bit far-fetched.