We report a detailed analysis of the cathodoluminescence spectra of ${\mathrm{Tm}}^{3+}$-implanted 2H-aluminum nitride (AlN) covering the wavelength range between 290 and $820\phantom{\rule{0.3em}{0ex}}\mathrm{nm}$ at temperatures between 12 and $60\phantom{\rule{0.3em}{0ex}}\mathrm{K}$. More than 200 transitions are observed, of which more than 100 of these transitions can be identified from emitting multiplet manifolds $^{1}I_{6}$, $^{1}D_{2}$, and $^{1}G_{4}$. Although the emitting levels are not observed directly, emission is also attributed to the $^{3}P_{2}$ and $^{3}P_{1}$ multiplet manifolds based on analyses of transitions to terminal levels $^{3}F_{4}$, $^{3}H_{5}$, and $^{3}F_{3}$. The observed crystal-field splitting of the ground-state multiplet manifold, $^{3}H_{6}$, and manifolds $^{3}F_{4}$, $^{3}H_{5}$, $^{3}H_{4}$, $^{3}F_{3}$, $^{3}F_{2}$, and $^{1}G_{4}$ is established from an analysis based on matching repeated energy differences between transitions. This method is similar to one used in analyzing arc and spark spectra. Temperature-dependent spectra also establish the crystal-field splitting of the $^{3}P_{1}$ and part of the manifold splitting of emitting levels such as $^{1}I_{6}$. To establish an initial set of crystal-field splitting parameters, ${B}_{\mathrm{nm}}$, that can be related to a physical model, we carried out a lattice-sum calculation by computing the crystal-field components, which are the coefficients in a multipolar expansion of the crystal field about the ${\mathrm{Al}}^{3+}$ sites that have ${C}_{3v}$ symmetry in the lattice. Emission channeling experiments indicate that the ${\mathrm{Al}}^{3+}$ sites serve as the substitutional sites for ${\mathrm{Tm}}^{3+}$ in AlN. With only minor adjustments to the calculated centroids to account for $J$-mixing, the calculated crystal-field splitting of most multiplet manifolds, $^{2S+1}L_{J}$, of ${\mathrm{Tm}}^{3+}(4{f}^{12})$ based on the ${B}_{\mathrm{nm}}$ obtained from the lattice-sum calculations, is in good agreement with the reported experimental splitting.