We present high-field magnetization and Br-79 nuclear quadrupole resonance (NQR) and Te-125 nuclear magnetic resonance (NMR) studies in the weakly coupled Cu2+ (S = 1/2) tetrahedral system Cu2Te2O5Br2. The field-induced level crossing effects were observed by the magnetization measurements in a long-ranged magnetically ordered state which was confirmed by a strong divergence of the spin-lattice relaxation rate T-1(-1) at T-0 = 13.5 K. In the paramagnetic state, T-1(-1) reveals an effective singlet-triplet spin gap much larger than that observed by static bulk measurements. Our results imply that the inter-and the intratetrahedral interactions compete, but at the same time they cooperate strengthening effectively the local intratetrahedral exchange couplings. We discuss that the unusual feature originates from the frustrated intertetrahedral interactions.