Antiferromagnetic insulators at low temperatures offer a clean arena to study non-semiclassical phenomena because the character of interactions is usually known. Macroscopic properties of the system can then be calculated by more or less sophisticated approximations and compared to experimental results. Generally, quantum effects are expected to increase by lowering the couplings or connectivity between the magnetic moments. Magnetic excitations, which can be easily measured by the neutron spectroscopy technique, are particularly sensitive to quantum effects. Answers to questions related to the ground state and the excitation spectra of quantum magnets are of primary importance to understand magnetism. We here study the coupled tetrahedra system Cu2Te2O5X2 (X=Cl, Br) and weakly connected Cu3TeO6 system. Both systems are complex magnetic insulators, with relatively low antiferromagnetic transition temperatures. In the Cu2Te2O5X2 system we observed an unusual situation where an anomalously weak low-energy Goldstone-like mode is accompanied by a strong higher energy gapped mode. When compared to a random phase approximation theory, there is a striking difference in the intensities, but also in the gap size. We propose that the origin of the discrepancy lies in the quantum fluctuations originating between the tetrahedra, which were not taken into account by the theory. In the "spin-web" lattice system Cu3TeO6, no reasonable fit of the semiclassical theory based on the Heisenberg Hamiltonian to the dispersion of excitations has been possible. Here, however, no quantum effects have been explicitly demonstrated. By doing additional polarized neutron spectroscopy experiment, we proved that a strong magnon-phonon coupling in this system significantly changes the properties of the spectrum.