Systematic studies of charge-dependent two- and three-particle correlations in Pb-Pb collisions at sNN = 2.76 and 5.02 TeV used to probe the Chiral Magnetic Effect (CME) are presented. These measurements are performed for charged particles in the pseudorapidity (η) and transverse momentum (pT) ranges |η| < 0.8 and 0.2 < pT< 5 GeV/c. A significant charge-dependent signal that becomes more pronounced for peripheral collisions is reported for the CME-sensitive correlators γ1, 1 = 〈cos(φα + φβ − 2Ψ2)〉 and γ1, − 3 = 〈cos(φα − 3φβ + 2Ψ2)〉. The results are used to estimate the contribution of background effects, associated with local charge conservation coupled to anisotropic flow modulations, to measurements of the CME. A blast-wave parametrisation that incorporates local charge conservation tuned to reproduce the centrality dependent background effects is not able to fully describe the measured γ1,1. Finally, the charge and centrality dependence of mixed-harmonics three-particle correlations, of the form γ1, 2 = 〈cos(φα + 2φβ − 3Ψ3)〉, which are insensitive to the CME signal, verify again that background contributions dominate the measurement of γ1,1.