Motivated by an emerging need for pertinent sizing and indexing of various data structures that are used for the efficient storage and processing of the reachability graph of certain bounded PN subclasses, this work investigates the complexity of the cardinality assessment of various marking sets that have been proposed as reasonable (over-) approximations for the set of reachable markings. Along these lines, our main results establish the # P-hardness of the aforementioned estimation for the most prominent of these marking sets. To the best of our knowledge, this is also a first attempt to provide formal # P-hardness results for counting problems that arise in the PN (and the broader DES) modeling framework.