Multiple disruptions in road networks have the potential for cascading effects that can cause significant degradations of network performance resulting in large increases in travel time. However, it is challenging to identify vulnerable combinations of links in road networks due to the complex interdependency of road segments and a prohibitively large number of possible combinations of links. In this paper, we present a deep reinforcement learning (DRL) framework to identify vulnerable combinations of links in road networks. We let a DRL agent select links to disrupt, and its policy is parameterized with deep neural networks (DNNs). The policy is directly learned from the consequences of disruptions of links (i.e., congestion incurred by the disruptions) in traffic simulations where multiple links are disrupted in a sequence. As a case study, we analyzed vulnerable combinations of links in the road network in the city of Davis in California, and compared the criticality of the disruptions of links selected by the proposed DRL-based method and heuristic-based methods that use betweenness centrality or traffic counts of links. In the results, we observed that disruptions by the DRL agent induced significantly larger congestion and increase in travel time of vehicles than heuristic-based methods. Furthermore, the links selected by the DRL agent reveal that the disrupted links are selected considering both the static properties such as the topology of the road network and the capacity of the links as well as dynamic properties imposed by the traffic demand.