Network feedback in a congestion-control system is subject to delay, which can significantly affect stability and performance of the entire system. While most existing stability conditions explicitly depend on delay D-i of individual flow i, a recent study [1] shows that the combination of a symmetric Jacobian A and condition rho(A) < 1 guarantees local stability of the system regardless of Di. However, the requirement of symmetry is very conservative and no further results have been obtained beyond this point. In this technical note, we proceed in this direction and gain a better understanding of conditions under which congestion-control systems can achieve delay-independent stability. Towards this end, we first prove that if Jacobian matrix A satisfies parallel to A parallel to < 1 for any monotonic induced matrix norm parallel to.parallel to, the system is locally stable under arbitrary diagonal delay D-i. We then derive a more generic result and prove that delay-independent stability is guaranteed as long as A is Schur diagonally stable [2], which is also observed to be a necessary condition in simulations. Utilizing these results, we identify several classes of well-known matrices that are stable under diagonal delays if rho(A) < 1 and prove stability of MKC [1] with arbitrary parameters alpha(i) and beta(i).