We develop probabilistic upper bounds for the matrix two-norm, the largest singular value. These bounds, which are true upper bounds with a user-chosen high probability, are derived with a number of different polynomials that implicitly arise in the Lanczos bidiagonalization process. Since these polynomials are adaptively generated, the bounds typically give very good results. They can be computed efficiently. Together with an approximation that is a guaranteed lower bound, this may result in a small probabilistic interval for the matrix norm of large matrices within a fraction of a second.