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 (p_T) ranges |η| < 0.8 and 0.2 < p_T< 5 GeV/c. A significant charge-dependent signal that becomes more pronounced for peripheral collisions is reported for the CME-sensitive correlators γ_{1, 1} = 〈cos(φ_α + φ_β − 2Ψ₂)〉 and γ_{1, − 3} = 〈cos(φ_α − 3φ_β + 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 γ_1,1. Finally, the charge and centrality dependence of mixed-harmonics three-particle correlations, of the form γ_{1, 2} = 〈cos(φ_α + 2φ_β − 3Ψ₃)〉, which are insensitive to the CME signal, verify again that background contributions dominate the measurement of γ_1,1. [Figure not available: see fulltext.]