The Wall Street Journal editorial page called it the “Throckmorton thumping.” Utah Republican Congressman Chris Cannon, a supporter of guest workers and “comprehensive immigration reform,” beat primary challenger Matt Throckmorton—“who made immigration the central issue of the campaign”by 16 points.

According to the Journal, the outcome was “noteworthy because national anti-immigrant activists—a motley band of population-control zealots and nativists—were hoping to make an example of Mr. Cannon.”

That was in 2004. Two years later, Cannon lost at the GOP state convention but held onto his House seat by winning the primary. Finally, in 2008 he was actually unseated by an anti-amnesty challenger, current Congressman Jason Chaffetz, who won 60 percent of the vote in the primary. No “Cannon thumping” editorials ensued.

This trip down Republican memory lane seems necessary in light of the commentary following House Majority Leader Eric Cantor’s shocking defeat in last Tuesday’s Republican primary. When even timorous supporters of amnesty for illegal immigrants win elections, it is said to be a validation of the winner’s immigration stance. When they lose or do worse than expected, immigration is said to be irrelevant to the result.

So it is unsurprising to see the conventional wisdom emerging that immigration had virtually nothing to do with Cantor’s loss, even though the winner Dave Brat ran hard against amnesty. A Public Policy Polling survey purports to show immigration was essentially a non-issue. The Atlantic contrasted Cantor with victorious South Carolina Sen. Lindsey Graham, known to his detractors as “Grahamnesty”: “Graham ran on immigration, while Cantor ran away from it.”

First, even if you believe pollsters can successfully recreate the Cantor-Brat primary electorate after the fact when so few of them correctly predicted what it would look like at the outset—this isn’t an exit poll—such polling tends to be flawed. The wording frequently gives respondents choices that don’t accurately reflect the debate actually in play and assumes much of what is in doubt. (That’s definitely true of the Public Policy Polling survey, commissioned by the liberal Americans United for Change.)

Most self-described amnesty opponents don’t believe that either the Obama administration or recent immigration legislation will actually meet the conditions said to differentiate “earned” legal status from amnesty. Most of them don’t view mass deportations as their preferred policy outcome. And many would be open to some form of legalization after a sustained, measurable reduction in the illegal immigrant population is achieved.

“The question isn’t whether we should grant unauthorized immigrants some kind of legal status; the question is when we should do that and what other policy changes should go along with it,” writes the policy journalist Robert VerBruggen in a more extensive criticism of recent immigration polling.

Graham’s showing in South Carolina is consistent with Cannon’s in Utah four years before he ultimately lost to an anti-amnesty challenge (in fact, it’s slightly worse): 44 percent of Republican primary voters cast ballots against him. Immigration was part of a cluster of issues motivating them.

Note that virtually nobody running in a Republican primary accepts the characterization of their immigration plans as amnesty, including not only Graham but also candidates as safe from conservative challengers as George W. Bush in 2004. Note that Graham floated a constitutional amendment reforming birthright citizenship when he was trying to sell his immigration plans to conservatives he acknowledged weren’t pleased with them and himself balked at advancing a bipartisan bill when doing so would have been injurious to his friend John McCain’s reelection prospects.

That year, McCain flipped on immigration in fairly dramatic fashion to win a Republican primary against anti-amnesty foe J.D. Hayworth. This was a repeat of when he pledged to oppose his own immigration bill in order to win the party’s presidential nomination—a bill that nearly tanked his campaign but was, fortunately for Arizona’s senior senator, a distant memory by the time the primaries rolled around.

Acquiring the pro-amnesty label is pretty clearly a net negative for most Republicans during the primaries. That’s why underfunded anti-amnesty challengers without much else going for them routinely break 40 percent against established GOP incumbents like Cannon, Graham, Renee Ellmers, and Jeff Flake.

Immigration is usually not enough to take down an incumbent by itself, however. Think of it as the political equivalent of an illness that only kills people with weakened immune symptoms. Perceived immigration impurity is even an obstacle that must be overcome at the presidential level, as McCain did and Rick Perry didn’t. (The jury is still out on Marco Rubio.)

Cantor had the underlying illness of not being terribly well liked in his district. Brat’s campaign was able to use the concern that the majority leader was about to double-cross grassroots conservatives on immigration to drive free media from the likes of Laura Ingraham, Mark Levin, and Ann Coulter (not to mention Mickey Kaus).

The challenger smartly tucked his anti-amnesty stance into a broader critique of the two parties’ fealty to corporate interests at the expense of ordinary Americans and excelled at the kind of retail politics Cantor bizarrely eschewed. (Sean Trende’s analysis of all this is hard to beat.)

None of Graham’s six challengers succeeded in distinguishing themselves from each other, much less the incumbent. (State Sen. Lee Bright came the closest.) Thus none were in the position to benefit from the kind of earned media a Levin or Ingraham could provide. Even if a challenger had broken away from the pack, Graham might still have won—an anti-amnesty candidate backed by talk radio failed to force a runoff in neighboring North Carolina and a cash advantage goes a much longer way in a statewide race—but a Brat-like upset would have at least become theoretically possible.

House Republicans currently seem poised to replace Cantor in the leadership with someone with similar immigration views. If history is any guide, amnesty skeptics need to persevere through a few thumpings—and benefit from a little luck—to come out ahead.

W. James Antle III is editor of the Daily Caller News Foundation and author of Devouring Freedom: Can Big Government Ever Be Stopped?