The damping coefficient of the power take-off (PTO) system is a key parameter in the performance assessment of a wave energy converter (WEC). However, since in most WEC studies the focus is mainly on the absorbed power, damping estimation is generally overlooked on the assumption that a single constant coefficient can properly characterize the WEC's damping of a given configuration for all wave conditions. Recently, while analyzing the experimental tests of CECO, a floating-point absorber WEC, significant discrepancies were found among their experimental responses under different incident waves. Instead of attributing those differences to nonlinear hydrostatic or Froude-Krylov effects, it was hypothesized that variations in the PTO damping associated to incident waves was the main cause. This study presents the experimental evidences of that behavior for regular and irregular waves. Furthermore, a hybrid approach for the assessment of damping coefficients is proposed and applied to CECO's experimental responses. The results demonstrated that: a) damping coefficients were significantly affected by wave conditions; b) higher PTO damping coefficients were obtained for milder irregular waves than for rougher regular waves; c) the hybrid approach reliably and efficiently estimated the WEC power in regular and irregular waves.