Determination of the optimal operating conditions for large-scale production of carbonaceous nanomaterials (CNTs), mainly carbon nanotubes (CNTs) but also graphene-related materials (GRMs), remains as a major industrial and scientific challenge. For this purpose, our group is conducting an extensive study to evaluate the effect of the reaction conditions and catalyst compositions on the productivity and selectivity to different carbonaceous nanomaterials. In the case of Co-based catalysts, we have found that the promotion with Cu or Mn has a dramatic effect on the type of carbon nanomaterials synthesized. Thus, the presence of Cu promotes the growth of CNTs, while Mn favors the formation of two-dimensional GRMs. In this context, we present here a comprehensive study of the CNTs growth via catalytic decomposition of methane using a Co-Cu/cellulose derived carbon catalyst. The influence of the reaction temperature (650–950 °C) and feed composition (7.1%–42.9% of CH4 and H2) on the yield and CNTs quality was evaluated. A transition in the characteristics of the carbonaceous nanomaterial growth was observed at about 800 °C. Below this temperature, the reaction was selective towards the formation of CNTs, while above 800 °C, the obtained nanomaterial exhibited a graphite-like morphology. In addition, the catalyst deactivation was quite low in the CNTs growth regime, attaining high productivity under 11 h of operation. The model used to study the kinetics of carbon formation allowed understanding the most influential variables in the growth process, revealing the existence of a transition temperature at which there is a change in the preferential path for the formation of CNTs or GRMs. After exploring a large set of reaction conditions, the best operating parameters for growing CNTs with high productivity (0.29 gC/gcat∙h) and quality (ID/IG = 1.10, I2D/IG = 0.13) were found at 750 °C under 28.6% CH4:14.3% H2:57.1% N2.