We investigate the issue of determining whether the intersection of a context-free language (CFL) and a Petri net language (PNL) is empty. Our contribution to solve this long-standing problem which relates, for instance, to the reachability analysis of recursive programs over unbounded data domain, is to identify a class of CFLs called the finite- index CFLs for which the problem is decidable. The k-index approximation of a CFL is obtained by discarding all the words that cannot be derived within a budget k on the number of non-terminal symbols. A finite-index CFL is thus a CFL which coincides with its k-index approximation for some k. We decide whether the intersection of a finite index CFL and a PNL is empty by reducing it to the reachability problem of a Petri nets with ordered inhibitor arcs, a class of infinite state systems for which reachability is known to be decidable. Conversely, we show that the reachability problem for a Petri net with ordered inhibitor arcs reduces to the emptiness problem of a finite-index CFL intersected with a PNL.