The New York Attorney General’s office said preliminary results of an investigation revealed “troubling” evidence of Time Warner Cable’s “apparent failure” to deliver New York customers the internet speeds it promises.

Tim Wu, special advisor to the attorney general, sent a letter to Charter Communications Inc., which recently acquired Time Warner Cable for $60 billion, saying he hopes Charter will “substantially improve the reliability, performance, and speed of the Internet delivered to customers, as well as how Time Warner...