We demonstrate for the first time, to the best of our knowledge, a thulium-doped, all-fiber, mode-locked laser using a carbon nanotube saturable absorber, operating in the dissipative-soliton regime and the stretched-pulse-soliton regime. The net dispersion of the laser cavity is adjusted by inserting different lengths of normal dispersion fiber, resulting in different mode-locking regimes. These results could serve as a foundation for the optimization of mode-locked fiber-laser cavity design at the 2 μm wavelength region.