The paper is mainly concerned with the ''true interval of orthogonality for a sequence of orthogonal polynomials, defined as the smallest closed interval containing the limit points of the set of zeros of these polynomials [cf. e.g. \textit{T. S. Chihara}, An introduction to orthogonal polynomials (1978; Zbl 0389.33008)]. Bounds are given for the endpoints of the interval in question, which are based upon an oscillation theorem of independent interest. Note that the author has shown quite recently that the results apply also to certain classes of stochastic processes.