Ezra Klein looks at Gallup's graph about public opinion over time and wonders why opinion about access and cost is so volatile, despite the actual situation not changing much over the time span covered.
While I don't have a specific answer, I think such an answer would lie in what we know about public opinion: it's not well informed, and it follows opinion leaders.
First of all, Gallup's question strikes me as an odd one. The wording is: "What would you say is the most urgent health problem facing this country at the present time?" Not surprisingly, the responses are a mix of diseases and public policy. Cancer, for example, is the response from 12% of those surveyed; obesity (a disease? a public policy problem?) is at 14%. So as diseases move in and out of the headlines, they'll leave more room for policy problems.
That's especially true because the question asks only about the (single) most urgent health problem. If something spikes up, everything else necessarily falls. So it's possible that the downward spike for access in the middle of the last decade may be the result of the bird flu scare, and the gradual rise of public policy issues in general may be because of the decline of HIV/AIDS as a relevant issue.
Then, we get to opinion leadership. My guess -- and again, this is just a guess -- is that if you looked at the rhetoric of Democratic politicians, you would find that they started using the word "access" in the late 1990s, and that the rise of access as the most urgent health problem was the result.
Hmmm...how about a little empirical research. The word "access" was used in the 1992 Democratic platform, but it doesn't show up at all in Bill Clinton's convention speech or his 1993 health care speech to Congress. In 2000, "access" is a bit more prominent in the platform, with the platform section entitled "Accessible, Affordable, Quality Health Care," compared to 1992's "Affordable Health Care," but it isn't in Al Gore's acceptance speech. By 2004, however, John Kerry wants health care that is "affordable and accessible" in his convention speech. Now, that's just a quick look -- to really do this, we would want to see word usage for ads and other candidate and surrogate speeches -- but for what it's worth it matches the increase in people reporting "access" as the most urgent problem from 1992 through 2004.
The story behind this is that most people, most of the time, don't think about public policy questions surrounding health care. They do think about their own health care, and about health care stories highlighted in the press, but not about policy issues. So it's not surprising at all that answers jump around a lot, probably driven by short-term changes in news coverage and political rhetoric.