The cartoons on view have a range of styles, sensibilities and some unexpected bursts of color: One from Mitra Farmand shows three protesters holding blank placards as they shout, “We need markers!”; Lars Kenseth considers a different meaning of the roadway merging of automobiles; and Amy Kurzweil depicts Mr. and Mrs. Potato Head playing Twister in a way only they can. Each of the cartoonists has had work published in the magazine; Mr. Kenseth has sold 12 cartoons for print and six for The New Yorker’s Daily Cartoons and Daily Shouts online sections. (Mr. Ostow has also had success there with two humorous illustrations for Daily Shouts.)

Julia Suits, who is based in Austin, Tex., has had about 40 cartoons published since 2005. And she said the rejection is not necessarily final, noting that illustrators are often encouraged to refine their work. Her three cartoons in the exhibition are at different stages of development, including one in which a mother’s summer camp send-off to her child does not feel like it’s “naturally falling out of someone’s mouth,” she said. She’s surprised the magazine has not taken one that stems from a word play on Möbius strip and Moby-Dick.

Her white whale, however, may be her cartoon about a family of balloons contemplating a circumcision. “It started out with two parent balloons, the baby balloon and a human surgeon,” Ms. Suits recalled. The original caption was, “Don’t worry, I’ve done hundreds of these.” Then the adjustments began. The current iteration has the balloon parents looking over their son and the caption, “I’m nervous about the circumcision, Gary.”

“The most overthought cartoons are the least successful,” Ms. Suits said. “The ones that pop out of nowhere are the ones that seem to work.”

Hilary Fitzgerald Campbell sold a cartoon in May, but it has not been published yet. “I’ve asked other cartoonists,” she said. “It normally takes months. I’m trying to be patient.”