Hard-carbon materials were made either by one-step or multi-step pyrolysis of cotton cloth between 700 and 1100 degrees C. All carbons have been characterized by gas sorption, X-ray diffraction (XRD) and small-angle X-ray scattering (SAXS) techniques. Two types of carbons have been obtained. One, made by multi-step pyrolysis, has the highest lithium reversible capacity [about 600 (mA h)/g] and two distinct voltage regions: a sloping one between 1.5 and about 0.1 V, called the high-voltage region (HVR), and a horizontal one between 0.1 and 0.1 V, called the low-voltage plateau (LVP). The other carbons made by the one-step process have only the HVR and less capacity [up to 470 (mA h)/g]. The influence of the current density and temperature on the capacity and degradation rate in both LVP and HVR was checked. We suggest that there are two different modes of lithium insertion: intercalation-like (on both sides of single graphene sheets) at lower potentials and chemical binding to edge carbon atoms at higher potentials vs. lithium reference electrode. A schematic model for lithiated carbon is proposed.