The easiest scenario would probably be for a military provisional government to assume temporary power after limited nuclear war destroyed the civilian government beyond repair within the existing constitutional framework, but that probably violates the spirit of the ACH.
Another category of scenario would be a counter-coup against a President who's executed a self-coup. Maybe the President has asserted extreme unconstitutional powers, somehow brushed aside attempts by Congress and the Courts to push back, and assembled enough of a clique in the Senate to stave off impeachment. The military then steps in to remove the President and holds power until they can sort things out.
Then there's
Seven Days in May scenarios, where military leadership with the ability to execute a coup believes there's an imminent existential threat to the US, which the President refused to do anything about. I think these have been discussed here in the past, and agreed to be plausible enough for a good story, but pretty darn unlikely to have actually happened.
Then there's a more traditional military coup scenario. Here's an article outlining a then-future-history scenario to that effect, written by a senior military officer in the early 90s:
http://strategicstudiesinstitute.army.mil/pubs/parameters/Articles/2010winter/Dunlap_Jr.pdf