It is well-known that the global asymptotic stability analysis of neutral systems is an important concept in designing the appropriate controllers or filters for this class of systems. This paper carries out a delay-independent stability analysis of neutral systems possessing discrete time delays in the states and discrete neutral delays in the time derivative of the states in the presence of nonlinear disturbances. Some new global asymptotic stability criteria are proposed by introducing a novel Lyapunov functional. The obtained delay-independent stability criteria establish some simple and easily verifiable mathematical expressions involving the elements of the system matrices and the disturbance parameters of the neutral system. Different from the most of the previously reported stability results for neutral systems, the conditions obtained in this paper are not expressed in terms the Linear Matrix Inequalities (LMIs). Therefore, the criteria presented in this paper can be considered as the alternative results to previously published stability results stated in the LMI forms. A comparison between the results of this paper and some of previously published corresponding stability results is made to substantiate the significant improvement of the proposed results. A constructive numerical example is also presented to show applicability and the effectiveness of the proposed stability condition.