What is drawing so many churners, travelers and others to the Sapphire Reserve card are its perks: Cardholders who spend $4,000 in the first three months get a sign-up bonus of 100,000 points, worth $1,500 in travel through Chase's website—and potentially more, some bloggers point out, when transferred to Chase travel partner sites. The card also gives three points for every $1 spent on dining and travel, and its definition of travel—which includes ride-share services like Uber and home-rental services like Airbnb—is broader than other cards'. It also offers a $300 annual credit to reimburse cardholders for travel expenses.

"Once people started finding out about this stuff, it started going crazy" online, Coomer said.

Though both he and his wife applied, he warned against getting too caught up in the enthusiasm online, saying the card might not be right for everyone. Its $450 annual fee raises the stakes for cardholders: That $1,500 in travel rewards isn't the same as $1,500 in cash, and you don't want to pay a hefty annual fee just for travel points you'll never be able to use.

Credit-card companies use generous perks to lure customers who travel frequently and spend freely, but they're wary of churners aiming mainly for free trips and other goodies. The companies are increasingly cracking down on the most hardcore churners, who take out card after card for their sign-up bonuses. Chase's new card application explicitly warns them away: "You will not be approved for this card if you have opened 5 or more bank cards in the last 24 months."

That blocks Dan Miller, a blogger and churner in Cincinnati, from getting one. He's disappointed, but he's philosophical.

"Part of what I enjoy about this is not only traveling," said Miller, who has turned credit-card rewards into free trips to Seattle, Singapore, Dubai and Lake Tahoe.

"This is a game," he said. "The rules change; you have to adapt."

As for the run on the Chase Sapphire Reserve cards? Customers who get the temporary plastic cards can use them as long as they want—but once the metal cards are available again, they can always upgrade.

Having that heavier, fancier card "does make a difference to people," said Leppar, who has one. "The whole purpose of it is to show it's a premier card."

This article was provided by Bloomberg News.

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