This paper considers some univariate and multivariate operational risk models, in which the loss severities are modeled by some weakly tail dependent and heavy-tailed positive random variables, and the loss frequency processes are some general counting processes. We derive some limit behaviors for the value-at-risk and conditional tail expectation of aggregate operational risks in such models. The methodology is based on capital approximation within the Basel II/III framework (the so-called loss distribution approach). We also conduct some simulation studies to check the accuracy of our approximations and the (in)sensitivity due to different dependence structures or to the heavy-tailedness of the severities.