This research evaluates the anaerobic digestion (AD) process of the residue generated in a new olive-oil manufacturing process for cold-pressed olive, a residue consisting of a mixture of the wastewater and solid waste obtained from this process. Additionally, in order to assess the possible influence of the level of ripening of the olives on the performance of anaerobic processing, olives of the Picual variety were collected at two stages, i.e., green olives and olives in veraison. The AD processes of the residues obtained from the cold-pressing process and the process without pressure (control) were comparatively assessed by means of biochemical methane potential (BMP) assays conducted at mesophilic temperature (35 ± 1 °C). Maximum values for methane yield (390 ± 1 NL CH4/kg VSadded) and biodegradability (84.5%) were obtained from the cold-pressed green olive residues. For the rest of the wastes studied, biodegradability also reached high values, ranging from 79.1 to 79.6%. The logistic model adequately fit the experimental data and allowed for the assessment of the anaerobic biodegradation of these wastes and for obtaining the kinetic parameters for each case studied. The theoretical values for ultimate methane production predicted from this model showed less than a 1% deviation from the experimental values. A decrease was detected for both types of olives tested in the rate of maximum methane production, Rm, during the cold-pressing process, from 44.3 ± 0.1 to 30.1 ± 1.3 L CH4/(kg VS·d) (green olives) and from 43.9 ± 1.5 to 38.7 ± 1.6 L CH4/(kg VS·d) (olives in veraison). Finally, the highest energy output result was detected in the waste from cold-pressed green olives (15.7 kJ/g VSremoved), which coincided with its high methane yield.