Yes — both names are British impositions on earlier Dutch and Indigenous designations.
1624: Dutch West India Company founds New Amsterdam on Manhattan Island (capital of New Netherland). 1664: English fleet (Richard Nicolls) demands surrender. Peter Stuyvesant capitulates. Renamed New York for James, Duke of York (future James II/VII), brother of Charles II. New Netherland → Province of New York.
Pre-colonial: Narragansett name Aquidneck ("island" / "place of the wild onion") for the island. 1636: Roger Williams founds Providence (mainland). 1638: Anne Hutchinson followers settle Aquidneck (Portsmouth, Newport). 1644: Parliamentary patent unites settlements as "Colony of Rhode Island and Providence Plantations."
Name origin debated:
Rhode Island is not an island (mostly mainland). "Providence Plantations" dropped from official name only in 2020 (voter referendum) due to slavery associations. New York's Dutch roots survive in Harlem (Haarlem), Brooklyn (Breukelen), Bronx (Bronck), Staten Island (Staaten Eylandt), Broadway (Brede Weg).