The inclusive energy distributions of complex fragments ($3\ensuremath{\leqslant}9$) emitted in the reactions $^{20}\mathrm{Ne}$ (145, 158, 200, 218 MeV) $+$ $^{27}\mathrm{Al}$ have been measured in the angular range ${10}^{\ifmmode^\circ\else\textdegree\fi{}}\text{\ensuremath{-}}{50}^{\ifmmode^\circ\else\textdegree\fi{}}$. The fusion-fission and the deep-inelastic components of the fragment yield have been extracted using multiple Gaussian functions from the experimental fragment energy spectra. The elemental yields of the fusion-fission component have been found to be fairly well explained in the framework of the standard statistical model. It is found that strong competition occurs between the fusion-fission and the deep-inelastic processes at these energies. The time scale of the deep-inelastic process was estimated to be typically in the range of $~{10}^{\ensuremath{-}21}$--${10}^{\ensuremath{-}22}$ s, and it was found to decrease with increasing fragment mass. The angular momentum dissipations in the fully energy damped deep-inelastic process have been estimated from the average energies of the deep-inelastic components of the fragment energy spectra. The estimated angular momentum dissipations, for lighter fragments in particular, are found to be greater than those predicted by the empirical sticking limit.