We describe an experiment in which Bose-Einstein condensates and cold atom clouds are held by a microscopic magnetic trap near a room-temperature metal wire 500 microm in diameter. The lifetime for atoms to remain in the microtrap is measured over a range of distances down to 27 microm from the surface of the metal. We observe the loss of atoms from the microtrap due to spin flips. These are induced by radio-frequency thermal fluctuations of the magnetic field near the surface, as predicted but not previously observed.