Ultrasmall double-barrier semiconductor structures are investigated in terms of the semiclassical sequential theory of resonant tunneling. The quantization of the charge buildup in the quantum well is taken into account. A peaked I-V characteristic is obtained, with each peak corresponding to an integer number of electrons in the well. A new explanation of the experiment of Reed et al. [Phys. Rev. Lett. 60, 535 (1988)] is proposed. The large value of the charging energy in their experiment ${\mathit{E}}_{\mathit{c}}$=${\mathit{e}}^{2}$/2C=43 meV makes semiconductor tunneling structures more convenient for observation of charging effects than the usual ones.