Regulatory CD4+CD25+ T cells play a major role in natural tolerance to body components and therefore are relevant to understand the self-non-self discrimination by the immune system. The most pressing theoretical question, regarding the fact that these regulatory cells perform their function through linked recognition of the APCs, is how this "non-specific" mechanism permits a proper balance between tolerance and immunity that is compatible with an effective self-non-self discrimination. To tackle this issue, we develop a numerical simulation, which extends a previous mathematical model of T-cell-mediated suppression to include the thymic generation and the peripheral dynamics of many T cell clones. This simulation can mimic the capacity of the immune system to establish natural tolerance to self-antigens and reliably mount immune responses to foreign antigens. Natural tolerance is based on ubiquitous and constitutive self-antigens, which select and sustain clones of specific regulatory cells. Immune responses to foreign antigens are only achieved if they displace the self-antigens from the APCs, leading to a loss of the regulatory cells, and/or if the foreign antigen introduction entails a sharp increase in the total number of APCs. Meaningful behavior is obtained even if differentiation of regulatory cells in the thymus is antigen non-specific, but requires that a minimum number of new T cells enter the periphery per unit of time, and that the repertoire is selected so that anti-self-affinities are within a proper interval. We conclude that positive selection is required to generate a sufficiently high frequency of self-antigen specific regulatory cells that reliably mediate natural tolerance. Negative selection is required to avoid the emergence at the periphery of very high affinity anti-self-regulatory cells that will make the tolerant state so robust that it could no be broken by the introduction of a foreign antigen. This result highlights the importance of repertoire selection in dominant tolerance proposing a novel role for the processes of positive and negative selection within this framework.