Some of the low-energy properties of the three-nucleon system are calculated with two s-wave spin-dependent interactions, which produce almost identical S13 and S01 two-nucleon phase shifts up to a lab energy of 350 MeV. Both interactions consist of an attractive square well and a short-range repulsion. The repulsive part of one potential is a hard core, for the other it is a nonlocal, separable potential. The hard-core potential produces a triton binding energy of 8.7 MeV, a doublet scattering length of 0.84 fm, and a quartet scattering length of 6.30 fm. The corresponding parameters calculated with the partly nonlocal potential are 8.8 MeV, 0.85 fm, and 6.31 fm. These results suggest that the low-energy properties of the three-nucleon system are insensitive to the details of the short-range part of the two-nucleon interaction.