That’s why movement in the polls has been particularly interesting lately. Over the last few months, online polls and live-interview polls have split in a way that would either support or undermine each of these theories, depending on which set of polls you believe.

The online polls support the polarization story: They show that the president’s approval rating has been astonishingly steady throughout his presidency, including over the last month.

But the telephone polls have been more variable, particularly over the last few months. In June and July, live-interview polls seemed to show the president’s approval rating matching the highest level of his term. If that’s accurate, it undermines the polarization theory and suggests the president still has the ability to broaden his appeal.

The president’s peak came in early July, just after the first Democratic debate, which could mean that the Democrats helped his cause, perhaps as a result of voters assessing his performance through the lens of the alternative. Democrats might have been particularly helpful to Mr. Trump if they repelled some persuadable voters by focusing on busing, ending private health insurance, extending health insurance to undocumented immigrants, or decriminalizing unauthorized entry into the United States.

Since then, the president’s ratings have dropped by several points, and there is no shortage of potential explanations: his verbal attacks on four congresswomen; mass shootings in El Paso and Dayton; growing concern about the economy and the trade war. A decline in the president’s ratings wouldn’t be good for his chances, of course, but it nonetheless suggests that public opinion is at least somewhat responsive to events, the economy and his conduct. If his ratings can go down, they can also go up.

It’s hard to know which set of polls is right. Live-interview polls have long been considered the gold standard in survey research. But they’re increasingly rare, which makes it harder to be sure of where they stand at any given time. They also aren’t typically weighted by a measure of partisanship, like party registration, so they’re more susceptible to shift with changes in which groups are likeliest to respond to polls.