Kate's penned a terrific post on the elemental confusion and complexity-of-use currently handicapping various low-income government health care programs. For instance, most folks thing Medicaid simply covers the poor. It does not. It covers certain categories of low-income mothers, all low-income pregnant women, some low-income men with families, etc, etc. It relies on more than 28 different categories of people, 21 of whom states have the option to cover (or not), and most of whom are never able to figure out where they fit or if they're eligible. It's a mess of a program that, while fairly efficient as a delivery structure, is poorly designed as a safety net.
But that's because Medicaid wasn't originally supposed to be a health system for low-income folks -- it was for widows, and the disabled, and other categories were added in piecemeal, inconsistent fashion. As for some of the other programs, check out the Healthy Families income chart. If you think that'd make sense to recent immigrants lacking a high school education, well, I applaud your belief in the mutability and adaptability of the human mind.
So yeah, yeah, you know I don't like our health structure. What to do about it? Well, basically, I agree with Kate's conclusions, so you should probably just read her whole thing.