A good example of this is Mitt Romney, whose ratings fell even lower than Mrs. Clinton’s during the Republican primary. He then faced a variety of attacks about his time at Bain Capital. He damaged himself with his “47 percent” comments. Yet despite it all, his ratings recovered by the 2012 general election. Republican-leaning voters, sure to vote against Mr. Obama, gradually figured out a way to like Mr. Romney. We saw the same thing with another Clinton, William Jefferson, who had even worse favorable ratings than Mrs. Clinton before quickly turning around his image once the 1992 general election began.

It’s useful to imagine a counterfactual: What if the economy had been stronger, and George H.W. Bush had had a 60 percent approval rating, on Election Day in 1992? Mr. Clinton wouldn’t have had a chance against those numbers, of course, but Mr. Bush wouldn’t have been running just on his record. He would have attacked Mr. Clinton’s character and he would have maintained a lead in the polls as a media narrative coalesced around a simple explanation: Mr. Clinton, with his weak numbers, was a smooth-talking womanizer who couldn’t be trusted. After his defeat, Mr. Clinton would have been remembered as yet another weak Democratic candidate in the footsteps of Walter Mondale and Michael Dukakis, fatally damaged before the campaign had even begun, rather than as one of the more talented politicians of our time.

For some, this line of thinking offers an uncomfortably deterministic theory about American presidential politics. But it helps explain a lot, such as why favorability ratings can move so much, even when there is relatively little new information about a candidate. It helps explain why, on the other hand, supposed game-changers often seem to have no effect. And it helps explain why, in the end, the polls move toward the fundamentals as the campaign evolves.

This doesn’t necessarily mean that Mrs. Clinton is in the clear. The fundamentals are strongest when an incumbent president is on the ballot because the feelings about the president are so strong and firm on either side. Mr. Romney, for instance, might have had a harder time improving his ratings and winning back G.O.P. voters — who disapproved of Mr. Obama — if his opponent had not been Mr. Obama.

For this cycle, the fitting analogy might be Al Gore, who started with far weaker favorability ratings than George W. Bush, with the backdrop of Mr. Clinton’s scandals. The economic fundamentals, though, seemed to suggest that Mr. Gore was in line to score a decisive victory. Indeed, the gap between Mr. Bush and Mr. Gore’s ratings closed as the race went on in 2000. But Mr. Gore, who like Mrs. Clinton was seeking to make it a third presidential victory in a row for Democrats, fell short.

More generally, there have been so few presidential elections in the era of modern polling that it would be a mistake to be too sure that Mrs. Clinton’s image won’t be a factor. In a century, political scientists could easily look back and say, with the help of new data, that favorability ratings can be decisive. For now, it would be unwise to rule it out. We already know that scandals can hurt incumbents in House elections, but it doesn’t come close to dooming them, studies have shown.

The issue with Mrs. Clinton’s email account and server doesn’t yet rise to the level of the scandals counted in the studies on House elections. But the possibility that the investigation will take a new turn can’t be ruled out. That’s the real danger for Mrs. Clinton, not her sagging ratings.