Following surgery and chemoradiation, ~50% of patients with locally advanced head and neck tumors experience relapse within the first two years, with a poor prognosis. Therefore, a novel therapeutic approach is required. The aim of the present study was to investigate the effect of combination treatment with the proteasome inhibitor bortezomib (BTZ), and ricolinostat (RCS), a specific inhibitor of histone deacetylase 6 (HDAC6), on CAL27 and Detroit562 head and neck cancer cells. BTZ and RCS exhibited cytotoxicity in a dose-and time-dependent manner. Simultaneous treatment with BTZ and RCS resulted in the synergistic enhancement of non-apoptotic cell death and autophagy. The receptor-interacting serine/threonine-protein kinase 1 (RIPK1) inhibitor, necrostatin, but not the autophagy inhibitor, 3-methyladenine, attenuated the cytotoxicity of combined BTZ and RCS treatment. Thus, necroptosis [type-III programmed cell death (PCD)], but not autophagic cell death (type-II PCD), appeared to contribute to the pronounced cytotoxicity. However, no phosphorylation of RIPK1 or mixed lineage kinase domain-like protein was detectable in response to BTZ or RCS. Furthermore, RCS induced α-tubulin acetylation and inhibited BTZ-induced aggresome formation along with endoplasmic reticulum stress loading. Combined treatment with BTZ and RCS enhanced the production of reactive oxygen species (ROS). The ROS scavenger, N-acetyl cysteine, abrogated the increase in cytotoxicity. These results suggest the potential therapeutic value of the dual targeting of the proteasome and HDCA6 for head and neck cancers through the induction of necroptosis-like cell death along with ROS generation.