Yet here’s one reason she might struggle to match Mr. Romney’s bump: His standing might have been a little deflated heading into the 2012 debate. Mr. Obama was just coming off his convention, and Mr. Romney was just coming off his remarks about the “47 percent.” You can also tell a similar story about the previous few debates: George W. Bush in 2000, John Kerry in 2004 and Barack Obama in 2008 were all challengers who fell behind after the incumbent party convention. They all gained after the first debate.

On the other hand, there are far more undecided voters and minor-party voters in 2016 than in 2012. Perhaps they’ll break for Mrs. Clinton.

But even if she does make big gains, it will be fair to wonder whether it actually matters. After all, Mr. Romney lost. In fact, he wound up losing by about the same margin he trailed by in the pre-debate polls: four points. The same story played out in 2000: Mr. Bush took a lead after the debates, and yet Mr. Gore ultimately came back and won the popular vote. In 2004, Mr. Kerry’s gains largely stuck, although he fell short in the election.

So what happened in 2000 and 2012? One possibility is that the shift in the polls was a bit of a mirage. The voters of the winning debater’s party became more excited, and therefore likelier to pick up the telephone — a phenomenon called partisan differential nonresponse. Or they became likelier to be counted by pollsters as likely voters. The reverse might happen to the losing side’s voters. These shifts might affect the polls, but might not have nearly as much of an effect on the underlying chance that voters actually turn out and support their candidates.

If Mrs. Clinton surges, it won’t be clear whether that’s a real shift in the race, even if it leaves Democrats feeling better.