The effect of DELTA and DELTA-DELTA degrees of freedom on the triton binding energy (E(T)) is studied using the Hannover DELTA-DELTA-force model. The three-body system of interest extends through J less-than-or-equal-to 4, with L(N DELTA),L(DELTA-DELTA) less-than-or-equal-to 4. A series of preliminary investigations reduces this three-body problem to J less-than-or-equal-to 2, with L(N DELTA),L(DELTA-DELTA) less-than-or-equal-to 2, and a 110 keV attractive correction to E(T). These J less-than-or-equal-to 2 DELTA-DELTA-calculations reveal a repulsive dispersive effect of 930 keV and an attractive DELTA-DELTA-three-body force effect of 500 keV, in addition to the corresponding one-DELTA-effects of 550 and 920 keV, respectively. The total DELTA-induced dispersive effect is thus about 1480 keV, while the total DELTA-induced three-body force effect is about 1420 keV: DELTA effects on E(T) almost exactly cancel. The net DELTA-DELTA J less-than-or-equal-to 2 result is E(T) = 7.32 MeV, while the corresponding nucleons-only result is 7.38 MeV. Similarly, the net J less-than-or-equal-to 4 result is E(T) = 7.43 MeV, as compared to the corresponding nucleons-only result, 7.46 MeV. The Hannover DELTA-DELTA-force model is also examined for consistency with the two-body scattering parameters and is found to be somewhat defective in this regard. Thus, the important implications of these qualitative results for nuclear physics are to some extent dependent on confirmation using more sophisticated force models.