The Min-$q$-Multiset Multicover problem presented in this paper is a special version of the Multiset Multicover problem. For a fixed positive integer $q$, we are given a finite ground set $J$, an integral demand for each element in $J$ and a collection of subsets of $J$. The task is to choose sets of the collection (multiple choices are allowed) such that each element in $J$ is covered at least as many times as specified by the demand of the element. In contrast to Multiset Multicover, in Min-$q$-Multiset Multicover each of the chosen subsets may only cover up to $q$ of its elements with multiple choices being allowed. Our main focus is a robust version of Min-$q$-Multiset Multicover, called Robust Min-$q$-Multiset Multicover, in which the demand of each element in $J$ may vary in a given interval with an additional budget constraint bounding the sum of the demands. Again, the task is to find a selection of subsets which is feasible for all admissible demands. We show that the non-robust version is NP-complete for $q$ greater than two, whereas the robust version is strongly NP-hard for any positive $q$. Furthermore, we present two solution approaches based on constraint generation and investigate the corresponding separation problems. We present computational results using randomly generated instances as well as instances emerging from the problem of locating emergency doctors.