A new stability criterion for time-varying systems consisting of linear and norm bounded nonlinear terms with uncertain time-varying delays is formulated. An explicit delay-independent sufficient stability condition is formulated in the terms of the transition matrix of the given linear part without delay and the bounds for the uncertain terms. The obtained condition turns out to be also necessary if the matrix of the linear part is time-invariant and symmetric; it is shown that these systems satisfy the well-known Aizerman's conjecture. The obtained criterion is contrasted by some of stability estimates available in the literature for these kinds of systems; in all cases the proposed criterion provides less conservative stability bounds.