The key with NEH is consultants. Like anything else the funding process is a political game, and getting scholars and consultants that are well known and established will help your chances greatly. Many, many films have been given a substantial boost by the NEH, and there's no reason that your film can't be boosted too.
The conclusion with NEH is that it's a good source for funding-- but the application process truly tests your mettle. It requires a lot of background work to be done before applications are even begun. The scope is broad, and the interests of the Endowment are far-reaching, but if you're not ready to make a full-blown film or don't have the resources to get started, you're better off looking at ITVS or PBS.
So what's the bottom line? Focus. The more focused you are about your project, the clearer your ideas will be when you present them, and the more likely you are to get funded. Accordingly, the more focused the funding endeavor, the more it reflects that the film is a labor of love, of passion. This is a quality that foundations have a hard time turning down when its apparent.If you're not completely focused yet, don't worry. There's still funding out there. These are simply examples of institutions whose very purpose is to fund projects. For prospective films or works-in-progress, your best bet is probably the private sector. For any film, though, going through every proposal process imaginable isn't a bad idea. Its not like you're not allowed to give it a try.