The statistical Hauser-Feshbach (HF) model performs poorly in calculating the ($\ensuremath{\gamma}$, $\ensuremath{\alpha}$) rates that are critical to the $p$ process. Experimental work on elastic scattering of the tellurium isotopic chain [A. Palumbo et al. (unpublished)] provided a new parametrization of the $\ensuremath{\alpha}$-optical potential and consequently new HF calculations of the ($\ensuremath{\alpha}$,$x$) cross sections on ${}^{120\ensuremath{-}130}$Te. However, reliable experimental cross sections of these isotopes have not been measured at energies relevant to the $p$ process. To test the reliability of the HF calculations, we measured the ($\ensuremath{\alpha}$,$n$) cross sections on ${}^{120}$Te, one of the $p$ nuclei, using the activation technique. The results are compared with the HF model calculations.