The oath to the constitution is a great example. That oath however, is there to ensure that you are bound not to a single person (president) but a way of life/national structure. If POTUS goes off the rails, declares martial law and abuses his powers, a case could be made that by the military standing by silently or refusing those orders... That you were upholding that oath. I guess it all depends on the circumstances.
I took a class years ago that briefly talked about this very subject. A legal scholar told us that in his opinion, the original 13 colonies retained the right to secede, because they were foreign entities first and states second. Other states however, were created by the US government buying the land they exist on, and as a result, could not. Course, that leaves Tejas in an interesting spot, because they were their own entity for a while.
Regardless, I think you'd see the nation break up into little kingdoms (states) in a large scale civil war. The national guard in those states could refuse their activations to Title 10 and serve the state instead if they chose. Talk about chaos. It would be like Turkey x10000.
Side note, there was talk a few months back about refusing to follow orders if the President demanded the military waterboard or torture prisoners. I guess the fundamental question is... at what point can the military say no?