The widely held notion that rising rents are making it nearly impossible to survive as a restaurant in the city received its ultimate endorsement this summer when celebrity restaurateur Danny Meyer said the Union Square Café would move from its longtime home on East 16th Street.

Mr. Meyer’s high-profile indictment of his rent bill played into the narrative of the struggling New York restaurant that has been amplified by a growing number of food blogs that chronicle every cafe and bistro’s closing in granular detail.

The numbers suggest, however, that the last decade has seen a boom for the industry, despite higher rents, a more demanding clientele and additional layers of red tape, including a letter-grade health-rating system from the city. The number of permits for restaurants, bars and cafes rose more than 27% to 23,705 at the start of fiscal year 2015 this July from 18,606 in fiscal year 2006, according to the city Department of Health.

Industry observers say more chefs looking to have their own place and dinner-party impresarios going pro have defied the odds and taken on rents that can climb above $15,000 a month in some Manhattan neighborhoods.

The Union Square Café had been in its location for nearly 30 years and is credited with helping to transform the once-dodgy neighborhood. A representative for Mr. Meyer declined to disclose the rent increase and said they planned to move when their lease is up at the end of 2015.

Last month, the owner of Angelica Kitchen, a vegetarian institution in the East Village, put out a plea to patrons to come often to help the restaurant stay afloat. The rent has risen to more than $22,000 a month. In its early days at a different location on St. Mark’s Place, the restaurant paid $450 a month. “How many more dragon bowls can I sell?” said owner Leslie McEachern.

September saw the closure of Yaffa Café, an East Village restaurant with simple food and a popular backyard. These are just a few high-profile examples from recent weeks, though actual numbers of closings are difficult to obtain. Some experts said many dining spots had fared poorly in part because they opened 15 or 20 years ago—the typical life of a lease—when rents were much lower.

“The ones that have been around that are very successful probably are surprised when their lease expires because they’ve had a nice run,” said Steven Spinola, president of the Real Estate Board of New York, a trade group.

Mr. Spinola said landlords had raised rents because of increased costs and increased demand for retail space—in part from more restaurants. “Higher rents are happening because there’s greater interest in opening up restaurants so there’s a greater demand on space,” he said.

Brooklyn, where retail rents are significantly lower than in Manhattan, and where the dining scene is light on haute cuisine and heavy on craft beer and homemade pickles, has seen the biggest percentage increase in new restaurants over the last five years.

Jaime Mendoza sets up tables in the dining room of the newly opened Le Fond in Greenpoint. Andrew Lamberson for The Wall Street Journal

The number of eating establishments in Brooklyn grew 10% to 5,658 at the start of fiscal year 2014, from 5,151 in 2009, according to the health department. Manhattan saw a 6% rise—to 9,654 establishments.

Veterans of New York’s unforgiving culinary scene question the staying power of the thousands of new dining spots.

“The restaurant industry is a high growth industry, but it is also a high turnover industry,” said Andrew Rigie, executive director of the New York City Hospitality Alliance, a trade group.

Drew Nieporent, the owner of Tribeca Grill and the Nobu chain of Japanese restaurants, said the phrase “fools rush in” came to mind when he looked at new restaurateurs. “If the rents were reasonable, I would have opened 20 more restaurants than I have today,” he said.

The numbers don’t indicate how much of the rise in permits has been driven by bars, casual cafes and takeout spots—which industry experts cite as the engine of the sector’s growth—as opposed to traditional sit-down restaurants.

Knowing the odds are stacked against new restaurants, Jake Eberle still decided to open a French bistro in Greenpoint two weeks ago. Mr. Eberle, 35 years old, a former chef de cuisine at the Lambs Club in Midtown, said it helped that rents in Greenpoint were about a third of even nearby Williamsburg. Le Fond has enjoyed a few glowing reviews in its early days.

“My father is a business owner, and I just always wanted to be my own boss and have control,” he said.

Still, Le Fond is likely to face challenges drawing clientele to a side street with few other restaurants. On a recent Tuesday evening—to be sure, a traditionally slow night in the restaurant business—it had about 10 customers at peak dining hour.