One of the most important antioxidant enzymes is superoxide dismutase (SOD), which catalyses the dismutation of superoxide radicals to hydrogen peroxide. The enzyme plays an important role in diseases like trisomy 21 and also in theories of the mechanisms of aging. But instead of being beneficial, intensified oxidative stress is associated with the increased expression of SOD and also studies on bacteria and transgenic animals show that high levels of SOD actually lead to increased lipid peroxidation and hypersensitivity to oxidative stress. Using mathematical models we investigate the question how overexpression of SOD can lead to increased oxidative stress, although it is an antioxidant enzyme. We consider the following possibilities that have been proposed in the literature: (i) Reaction of H2O2 with CuZnSOD leading to hydroxyl radical formation. (ii) Superoxide radicals might reduce membrane damage by acting as radical chain breaker. (iii) While detoxifying superoxide radicals SOD cycles between a reduced and oxidized state. At low superoxide levels the intermediates might interact with other redox partners and increase the superoxide reductase (SOR) activity of SOD. This short-circuiting of the SOD cycle could lead to an increased hydrogen peroxide production. We find that only one of the proposed mechanisms is under certain circumstances able to explain the increased oxidative stress caused by SOD. But furthermore we identified an additional mechanism that is of more general nature and might be a common basis for the experimental findings. We call it the alternative pathway mechanism.