During the 2360s, the USS Bradbury was still undergoing warp drive performance tests and serving as a transport to Earth from Betazed in late 2366. (TNG: "Ménage à Troi", "Brothers", "Chain of Command, Part I" production art)
The Bradbury-class was first named in the chart Starfleet Operations - Sectors 21166-23079 in "Brothers". According to the Star Trek Encyclopedia (3rd ed., p. 53) the Bradbury was named for s-f/fantasy writer, Ray Bradbury, a friend of the late Gene Roddenberry.
A ship description was mentioned in the supplements to the Last Unicorn Games role-playing games, which refer to this vessel as being a 335-meter-long heavy frigate in service from 2362, with ten decks, and describes it as being similar to the unfilmed Intrepid-class prototype designed during pre-production of Star Trek: Voyager.