We have found experimentally that the shot noise in InAlAs-InGaAs-InAlAs Triple-Barrier Resonant-Tunneling Diodes (TBRTD) is reduced over the 2eI Poissonian value whenever their differential conductance is positive, and is enhanced over 2eI when the differential conductance is negative. This behavior, although qualitatively similar to that found in double-barrier diodes, differs from it in important details. In TBRTDs the noise reduction is considerably larger than predicted by a semi-classical model, and the enhancement does not correlate with the strength of the negative differential conductance. These results suggest an incomplete understanding of the noise properties of multiple-barrier heterostructures.