Politicians have to be careful what they say. Sure, an ill-chosen word can be a career-ending gaffe, but even worse, the bully pulpit of elected office can help spread dangerous and misguided ideas. Chris Christie and Barack Obama are both in the news Monday for comments about vaccinations, but while Christie’s ham-handed remarks have absorbed most of the backlash, the president’s well-intentioned ones may be more perilous.

Sunday, in a pre-Super Bowl interview, President Obama was asked about the measles outbreak that apparently began at Disneyland. As of Saturday, the CDC reported more than 100 cases in 14 states. "The science is, you know, pretty indisputable," Obama told NBC's Savannah Guthrie. "We’ve looked at this again and again. There is every reason to get vaccinated, but there aren’t reasons to not .... You should get your kids vaccinated."

Monday, New Jersey Governor Chris Christie was asked about vaccines during a trip to England. Here's what he said:

We vaccinate ours [kids], and so, you know that’s the best expression I can give you of my opinion. You know it’s much more important what you think as a parent than what you think as a public official. And that’s what we do. But I also understand that parents need to have some measure of choice in things as well, so that’s the balance that the government has to decide.

Ironically, Christie was speaking outside a facility owned by an American company that manufactures a nasal flu vaccine. He is being derided as a "vaxer," but his statement is a little more nuanced than he's getting credit for. Though not nearly as strong as what Obama said, the governor made his own views clear. And after fierce backlash, he partially walked back his comments in a statement, saying he only meant to point out that states have discretion in which vaccines they mandate.

This isn't really a new position for Christie. In 2009, before he was elected, he told Don Imus he would give parents with concerns "a seat at the table." What is worrisome is that Christie cracked open the door to a debate where, as Obama says, the science is settled—it's a variation on the old "just asking questions" routine.

Christie is not the first nationally known Republican to flirt with the anti-vaccine movement—then-Representative Michele Bachmann infamously argued for a link between the HPV vaccine and mental retardation during and after a 2011 GOP presidential debate. But Bachmann has always been seen as a somewhat fringe figure, and in that case she was criticizing then-Texas Governor Rick Perry, no one's idea of a squishy liberal. Christie, on the other hand, is the somewhat more moderate governor of a northeastern state, a major presidential contender, and perhaps the highest-profile officeholder to countenance anti-vaccine theories. (As his national ambitions grow, Christie has tacked right on climate change as well.)

A world in which support or opposition to vaccination could become a partisan litmus test would be a dangerous one. It's not that hard to imagine—just look at climate change, once a relatively uncontroversial issue that has shifted to the point that Republican officeholders widely reject it. (A poll over the weekend suggested that rank-and-file Republican voters are far more open to admitting the reality of warming.)

So far, the partisan valence of vaccinations hasn't been especially clear. Anti-vaccination proponents are sometimes portrayed as oddball hippie types; worries about vaccines and genetically modified foods are portrayed as the left's answer to climate-change denialism. But a January 2014 paper by Harvard and Yale researchers found little correlation between left-right politics and vaccine skepticism.

Yet as my colleague Cari Romm reported in December, trying to bust the myths about vaccines causing autism or other cures can actually backfire. Research from Brendan Nyhan and Jason Reifler shows that beliefs about questions like vaccines are often not derived from empirical evidence. And trying to convince people out of their beliefs can actually encourage them to hold those views more staunchly, since they flow out of identity with a community rather than facts.

That means Obama's unequivocal statements Sunday might backfire, encouraging the anti-vaccine movement and more deeply entrenching beliefs. Maybe Christie will be an anomaly, or perhaps other Republicans will line up to agree. Phrasing it not as a question of scientific effects but as a matter of whether the government is overreaching is a somewhat innovative approach, and just the sort that could help to make vaccines a partisan issue for limited-government advocates. Unfortunately, following Nyhan and Reifler's research, Christie's "clarification" is likely to make much less public splash than his original statement.

It's a catch-22 for public-health officials: They can't sit idly by while vaccination numbers plunge and mostly vanquished diseases make a comeback, but anything they do to encourage vaccinations may instead encourage the opposite. One thing health officials could do without is politicians making their work harder.