While studying magnetism of d- and f-electron systems has been consistently an active research area in physics, chemistry, and biology, there is an increasing interest in the novel magnetism of p-electron systems, especially in graphene and graphene-derived nanostructures. Bulk graphite is diamagnetic in nature, however, graphene is known to exhibit either a paramagnetic response or weak ferromagnetic ordering. Although many groups have attributed this magnetism in graphene to defects or unintentional magnetic impurities, there is a lack of compelling evidence to pinpoint its origin. To resolve this issue, we systematically studied the influence of entropically necessary intrinsic defects (e.g., vacancies, edges) and extrinsic dopants (e.g., S-dopants) on the magnetic properties of graphene. We found that the saturation magnetization of graphene decreased upon sulfur doping suggesting that S-dopants demagnetize vacancies and edges. Our density functional theory calculations provide evidence for: (i) intrinsic defect demagnetization by the formation of covalent bonds between S-dopant and edges/vacancies concurring with the experimental results, and (ii) a net magnetization from only zig-zag edges, suggesting that the possible contradictory results on graphene magnetism in the literature could stem from different defect-types. Interestingly, we observed peculiar local maxima in the temperature dependent magnetizations that suggest the coexistence of different magnetic phases within the same graphene samples.