Resonant-tunneling diodes designed to have features in their current-density-voltage (J-V) characteristics in addition to the main peak often incorporate triple-barrier structures. In designing such structures, much attention is paid to the alignment of the quasibound levels in the two wells in order to achieve additional peaks or kinks in the J-V curve. Unfortunately, many such devices fail to display these additional features. It is commonly thought that this failure is solely due to the limitations of coherent tunneling models, but this is not always the case. Here we demonstrate that the simplest and most commonly employed approximation for the tunneling current density (the one-dimensional approximation) is often incorrect for triple-barrier devices and that when a more accurate approximation (the two-dimensional approximation) is used the J-V characteristics can be markedly different.