Through a listing on StreetEasy, they connected with Phillip Salem, a licensed agent at Triplemint.

“There was something so incredibly comforting in his voice,” Ms. Tatochenko said. “You can imagine how stressed I was.”

The couple was hoping to find three bedrooms — for them, the boys and the nanny — for less than $2 million. They wanted natural light, ample storage and space for “a table to do homework and chill out as a family,” Ms. Tatochenko said. And they needed a dog-friendly building for Lili, whom they adopted while living in Japan.

But “a three-bedroom apartment on the Upper East Side for under $2 million is like a needle in a haystack,” Mr. Salem said. Many apartments they saw were combinations of what had been one-bedrooms and studios in white-brick buildings dating to the early 1960s.

Image At the Rio, in Lenox Hill, a three-bedroom had 1,550 square feet, and the couple loved the pool. But the monthly charges of nearly $3,800 were uncomfortably high. Credit... Stefano Ukmar for The New York Times

In new buildings, prices were high and square footage low. “They think 1,200 is a good square footage for a three-bedroom, and it drives me nuts,” Ms. Tatochenko said. “You can’t fit into that with a proper family. I cannot stand clutter, and kids come with clutter, they do naturally.”

Size was a problem at the Rio, an amenity-rich condominium, circa 1987, in Lenox Hill, where a three-bedroom came with 1,550 square feet. The couple loved the pool and were indifferent to the wraparound balcony. The price was $1.83 million, with monthly charges of nearly $3,800 — uncomfortably high for them. It later sold for $1.685 million.

In Yorkville, one compelling apartment was in a 1930 co-op building, with a T-shaped layout resulting from a combination of two studios and a one-bedroom. The price was almost $1.6 million. But the maintenance, at more than $4,300 a month, was out of reach. That apartment remains on the market.