Indeed, the Indian Health Service (IHS) has an interesting history that could provide significant pointers for what may or may not be feasible for a national health care system.
As you state, the IHS system is underfunded. It has been funded a approximately 60% of need for well over a decade now. I'm not an expert in the causes/effects of that, but have some opinions on a broad scale.
The IHS system in Alaska was nearly non-functional until the early 1990's when the infrastructure and management was transfered to non-profit corporations operating under the sovereignty of Alaska Native Tribal governments. Since that time the effectiveness and the efficiency of the system has increased steadily, even though Congress underfunds. One reason is because the system is not actually single payer, and the various Native health corporations have found ways to bill other programs (Medicare/Medicaid) and to in some cases incorporate entire programs that are otherwise funded (grants etc).
While the system functions better today than it did when the Federal Government managed it, there are still huge cracks that are caused by the lack of 100% funding from Congress. For example, there is limited financial support for specialty clinics at various bush regional hospitals; one instance that I am aware of (painfully aware of) is that while Barrow is regularly visited by a cardiologist which no doubt helps reduce the number of deaths due to heart disease, there is never a urology clinic and therefore the prognosis for males who develop prostate cancer is almost certain to be poor (they move to Anchorage, die, or go bankrupt).
Both in Alaska and across the country statistics for many indicators (infant death mortality, longevity, incidence of diabetes, heart disease, cancer and others) indicate that the Indian Health program in the US is not a functional program at the Federal level, but rather a terrible embarrassment for America. In that light it is worth pointing out that Congress, in considering the current legislation for national health care, specifically requested testimony from representatives of the American Indian and Native Alaska tribal health organizations about how the new system would affect Indian health care programs. Their testimony was as much about how to avoid harming the Indian system as it was how to help it, and unfortunately it seems that nobody was asked how the AI/NA system could be used to improve health care on a national level.
I know of at least two representatives from Alaska that were asked to provide testimony to Congressional hearings (there may well have been others too). Evangelyn Dotomain, President/CEO of the Alaska Native Health Board gave testimony on June 25, 2009:
http://www.anhb.org/documents/ANHB+HNR+IHCIA+Testimony+Final1.pdfValerie Davidson, Senior Director of Legal and Intergovernmental Affairs, Alaska Native Tribal Health Consortium, testified on June 11, 2009:
http://indian.senate.gov/public/_files/ValerieDavidsontestimony.pdf