Interactive Markov Chains ($\text {IMCs}$) are compositional behaviour models extending both Continuous Time Markov Chain (CTMC) and Labeled Transition System (LTS). They are used as semantic models in different engineering contexts ranging from ultramodern satellite designs to industrial system-on-chip manufacturing. Different approximation algorithms have been proposed for model checking of $\text {IMC}$, with time bounded reachability probabilities playing a pivotal role. This paper addresses the accuracy and efficiency of approximating time bounded reachability probabilities in $\text {IMC}$, improving over the state-of-the-art in both efficiency of computation and tightness of approximation. Experimental evidence is provided by applying the new method on a case study.