The existence of inter-dependence between multiple networks imparts an additional scale of complexity to such systems often referred to as ‘network of networks’ (NONs).We have investigated the robustness of NONs torandom breakdown of their components, as well as targeted attacks, as a function of the relative proportion of intra and inter-dependence among the constituent networks. We focus on bi-layer networks with two layers comprisingdifferent numbers of nodes in general and where the ratio of intra-layer to inter-layer connections, r, can be varied, keeping the total number of nodes and overall connection density invariant. We observe that while the responsesof different networks to random breakdown of nodes are similar, dominantly intra-dependent networks ($r \ll 1$) are robust with respect to attacks that target nodes having the highest degree but when nodes are removed on the basis of the highest betweenness centrality (CB), they exhibit a sharp decrease in the size of the largest connected component (LCC) (resembling a first-order phase transition) followed by a more gradual decrease as more nodes are removed (akin to a second-order transition). As r is increased resulting in the network becoming strongly interdependent ($r \gg 1$), we observe that this hybrid nature of the transition in the size of the LCC in response to targeted node removal (based on the highest CB) changes to a purely continuous or second-order transition.We also explorethe role of layer size heterogeneity on robustness, finding that for a given r having layers comprising very different numbers of nodes results in a bimodal degree distribution. For dominantly inter-dependent networks, this resultsin the nodes of the smaller layer becoming structurally central. Selective removal of these nodes, which constitute a relatively small fraction of the network, leads to breakdown of the entire system – making the inter-dependent networks even more fragile to targeted attacks than scale-free networks having power-law degree distribution.