The New York attorney general’s office has opened a probe into whether Cablevision Systems Corp., Verizon Communications Inc. and Time Warner Cable Inc. are providing consumers with the Internet speeds they advertise.

In separate letters to all three providers, the attorney general’s office expressed concern that people aren’t getting the broadband speeds that companies are promising for reasons within the operators’ control. It also said consumers who are paying up for premium, superfast tiers may not be experiencing “proportional...