The long-time dynamics of modulated waves in a nonlinear LC transmission line is investigated. Considering the higher-order nonlinear Schrödinger equation, we define analytically the conditions leading to the instability of modulated waves. We show that two kinds of instabilities may develop in the network depending on the frequency range of the chosen carrier wave and on the magnitude of its initial amplitude, which is confirmed by our numerical simulations. The nonreproducibility of numerical experiments on modulated waves is also considered.