The manifestations of the symmetry-breaking artifact in three-electron-bonded systems have been investigated at several computational levels including second-order Moller-Plesset perturbation theory (MP2), coupled cluster (CC), and Bruckner-coupled cluster (B-CC) theories. The model systems, [H(n)X4XH(n)]+(X=Ne, F, O, N, Ar, Cl, S, P; n=0-3) cover all types of three-electron bonds that can possibly take place between atoms of the second and third rows of the Periodic Table. The critical interatomic distance beyond which symmetry breaking begins to take place at the Hartree-Fock and Moller-Plesset levels are determined for each model system. Their magnitude are found to obey regular tendencies which are related to the compactness of the orbitals involved in the three-electron bonds. In all model systems, the onsets of symmetry-breaking at the MP2 level are greater or equal to the equilibrium bonding distance between the XHn fragments. The symmetry-breaking artifact results in severe discontinuities in the dissociation curves at the MP2 level. The CC level pushes away the occurrence of the artifact to larger distances but do not remove the discontinuities. The artifact is practically cured at the B-CC level with perturbative treatment of triple excitations. The onset of symmetry-breaking may in some cases be shortened by substituent effects, to the extent that it becomes shorter than the equilibrium bonding distance like in the Me4O2+ and Me2F2+ cation radicals that are found to be symmetry-unstable even in their equilibrium geometries. The artifact carries over to unsymmetrical systems that display close functional resemblance to symmetrical systems, leading to convergence difficulties, erroneous geometries, and unphysical localization of the electronic charge. An economical alternative to the MP2 method, based on the average quadratic coupled-clusters (AQCC), is proposed for such cases, or in cases some stretched three-electron-bonded systems or full dissociation curves are to be investigated.