The superconducting property of Tl2Ba2CuO6+y (TBCO) with T(c) = 72, 40 and 0 K has been investigated by Cu-63 NQR and NMR and Tl-205 NMR measurements. The behaviors of the Knight shift and nuclear spin lattice relaxation time T1 below T(c) have been found to be almost the Same as those in YBa2Cu3O7, although the behavior above T(c) is strongly modified in the sense that the q-dependence of the antiferromagnetic (AF) spin fluctuation becomes less distinct upon doping. According to the argument for YBCO7 by Ishida et al., the T-dependence of 1/T1 and the Knight shift below T(c) in TBCO are well interpreted by a "gapless d-wave model" with a finite density of states at the Fermi surface. It is pointed out that impurities and imperfections and/or the applied magnetic field more or less cause the pair breaking in TBCO. Then the low-T properties in high-T(c) oxides are concluded to be seriously affected by the small but finite density of states at the Fermi level depending on the material.