It is certainly possible though if anything I only see it coming from the (if we're talking about the 30s) the so called American "Left". I think if for example, let's say Roosevelt dies, or looses because people see he's in a wheelchair (I never actually bought that he'd lose if they found out).
Then assuming a worse depression, the country becomes so destabilized then something is certainly possible. I think, maybe a split EC might do it. Let's Huey Long, and two moderate Republican and Democratic candidates. Than I could, assuming the depression is worse then OTL see a U.S military coup to back Huey Long into the White House, if the House decides against him (however to make this most realistic I'd suggest him getting a huge chunk of the PV and almost all the necessary EV, but just short). However, once the coup is done, Huey probably uses the military/National Guard to create some semi-dictatorship like he did in his home state. He wouldn't actually establish a full on dictatorship, the Army would remove him.
Another idea is perhaps during the civil war a plan like J. W. Booth's plan succeeds and almost the entire cabinet and some congressional leaders are killed/incapacitated. Then the army takes the reigns of the government ostensibly to finish the war and preserve democracy. However, the regime 'survives' through (after winning the war, or loosing it) using the suspension of Habeus Corpus and other similar measures to ensure they stay in power, though after a few generations I don't see this sort of statocracy remaining in place unless the Union looses the Civil War.