I think a third party could work, but it would need to have a strong regional basis. Your best shot for this would probably be FDR getting into trouble in the mid-1930s (or dying prematurely), giving the Farmer-Labor Party (MN), Progressive Party (WI), and the Nonpartisan League (ND) an opening. Though not a national presence, those three states would give enough of a geographic rump to keep a party going, and quite possibly build from.
Other than this, a catastrophic split within the Democratic Party would be your best bet...if the Dixiecrats walk out (I've toyed with a variation on this in a timeline), it's not likely that the GOP of the 1930s and 1940s would work with them directly (doing so would have been toxic in many regards) while the GOP had a bad name for historical reasons in most of the South, so I can see a situation emerging where the GOP and the National Democrats fight in most of the country, while the National Democrats and Dixiecrats fight in the South. Your only "real" three-way fights would be in the border/mountain states and in Texas. Basically, Duverger's Law still applies in most cases...you just have it applying on a sub-national level. Witness Canada (where you've ended up with various regions where one of the "big three" parties simply wound up being uncompetitive...the NDP, for example, never had much of a foothold in Quebec until recently, while the Liberals often had a hell of a time out on the Prairies) or Britain (where the Tories are basically locked out of Scotland, in particular, while Labour is more or less locked out south of the Thames) for ways this can work.
Structurally, I think that both the House and Senate could support a third party (so long as that party had a solid geographic base of some sort)...it's the Presidency that the party would likely be locked out of and which would make things difficult in the long run.