Ms. Wiley said that her staff was negotiating with Verizon and that the city would prefer not to sue the company for failing to fulfill its obligation. But, she added, “if that’s what we have to do, then that’s what we’ll do.”

Customers are still being told by Verizon that FiOS is “not available” to them, even after the company claimed to have wired the entire city. Ms. Brooks said that the last time she heard from Verizon was a letter in January telling her that the company was trying to figure out how to connect to her home.

One of Ms. Brooks’s neighbors, Barbara Cooke-Johnson, said that two years ago she escorted a Verizon representative door to door on her block, seeking permission to run fiber through the properties.

Ms. Cooke-Johnson, 75, a retired lieutenant for the city’s Emergency Medical Service, said she got about 10 neighbors to agree and thought that she would have upgraded from her poky DSL service by now. But she said that she never heard from Verizon again and remained stuck with an Internet connection so slow that it takes nearly 15 minutes to download her email.

Her 30-year-old grandson, Laquan, was particularly annoyed, she said, because “he’s on the Internet all the time.” Some neighbors complain that it is impossible for them to work from home without a fast, reliable connection to the Internet, Ms. Cooke-Johnson said.

Like most New Yorkers, they have been bombarded with ads for the FiOS service on television and, when they can get online, the Internet. Many of them have been persuaded, through those ads or personal experience, that FiOS is superior to competing services offered by the cable companies that had monopolies in different parts of the city before Verizon was granted a franchise to compete with them.