We report on the experimental studies of an interband tunneling effect between GaSb valence-band and InAs conduction-band quantum wells in a GaSb/AlSb/InAs/AlSb/GaSb/AlSb/InAs triple-barrier interband tunneling device. Multiple negative differential resistances were observed both at room temperature and 77 K. By varying the InAs well width to adjust the alignment of the conduction-band and the valence-band quasi-bound states, we observed more than one order of magnitude variation in the peak current density, indicating a significant quantization effect. Possible current conduction mechanisms were discussed based on the dependence of the peak-current density on the InAs well width.