And while it's tempting to dismiss these theories as rumor-mongering coming from fringe elements and people just spouting off, a new poll suggests that suspicions about her health are actually pretty widespread in the Republican Party.

The Gallup poll, in fact, shows just 27 percent of Republicans believe Clinton is "healthy enough to be president." And it's not just them; 63 percent of independents say this description applies to Clinton, meaning nearly 4 in 10 aren't convinced that it does.

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About 9 in 10 Democrats (89 percent) say she is healthy enough.

The poll didn't test whether these skeptics buy into specific conspiracy theories, mind you; perhaps they believe her pneumonia diagnosis was bad enough to disqualify her from serving as president. But that would be a pretty low bar for disqualification. And either way, it's still remarkable that nearly three-fourths of Republicans won't say she meets this very basic standard.

There is certainly some partisanship at work here, of course. On the flip side, just 54 percent of Democrats say Donald Trump is healthy enough to serve as president, despite no real indication that he struggles with a particularly serious health condition. Trump certainly hasn't been forthcoming with his health information — releasing a hyperbolic letter from his doctor last year and then appearing on Dr. Oz's show last week to disclose more — but there's little reason besides his age (70) to believe he's not in good health.

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Do all of the people who tell pollsters they harbor grave doubts about the health of a politician they dislike sincerely believe that politician is in poor medical condition? Not necessarily. Clearly, when asked about a candidate they oppose, many people are simply liable to say that candidate doesn't meet any qualification for being president.

But the flap over Clinton falling ill, and her campaign's tardy disclosure about it, has led to a renewed focus on candidate health. The same Gallup poll shows 51 percent think the candidates should release all of their medical information. That number is up from 38 percent in 2004, the last time Gallup asked this question.

That rise is driven by Republicans, 66 percent of whom now say a candidate should release everything. About half of independents and Democrats agree.