Thermodynamic experiments as well as Raman scattering have been used to study the magnetic instabilities in the spin-tetrahedra systems Cu(2)Te(2)O(5)X(2), X = Cl and Br. While the phase transition observed in the Cl system at T(N) = 18.2 K is consistent with 3D antiferromagnetic ordering, the phase transition at T(o) = 11.4 K in the Br system has several unusual features. We propose an explanation in terms of weakly coupled tetrahedra with a singlet-triplet gap and low-lying singlets.