In this paper, we achieved the output of switchable multi-wavelength mode-locked pulses delivering picosecond and femtosecond solitons simultaneously by means of adding a chirped fiber Bragg grating (CFBG) into an erbium-doped mode-locked fiber laser based on carbon nanotubes. By adjusting a polarization controller, we output the switchable multi-wavelength mode-locked pulses near 1530.8 nm, 1549.5 nm, and 1556.5 nm, with the pulse duration of 833 fs, 7.43 ps, and 899 fs, respectively. In addition, the formation of the above three wavelengths could be jointly attributed to the gain spectra of the erbium-doped fibers and the filtering effect of CFBG. In conclusion, the study provides a reference for designing the multi-wavelength mode-locked fiber lasers with different pulse duration.