With the increasing size and complexity of modern infrastructure networks rises the challenge of devising efficient and accurate methods for the reliability analysis of these systems. Special care must be taken in order to include any possible interdependencies between networks and to properly treat all uncertainties. This work presents a new approach for the reliability analysis of complex interconnected networks through Monte Carlo simulation and survival signature. Application of the survival signature is key in overcoming limitations imposed by classical analysis techniques and facilitating the inclusion of competing failure modes. The (inter)dependencies are modeled using vine copulas while the uncertainties are handled by applying probability boxes and imprecise copulas. The proposed method is tested on a complex scenario based on the IEEE reliability test system, proving its effectiveness and highlighting the ability to model complicated scenarios subject to a variety of dependent failure mechanisms.