Our nation is set up as a union of 50 separate States, plus one Federal district, which have agreed to have one Federal law be soverign over the laws of the individual States. Nevertheless, those 50 States are set up with governments independent of one another. As such, each state would still be able to function even if another state's government were to fail, or even if the Federal government were to fail. Even if, say, something catastrophic were to happen in Mississippi, that would not cause a domino effect of collapse across the rest of the nation.
This even applies to elections.