Existential rules are a prominent formalism to enrich a database with knowledge from the domain of interest, but make even basic reasoning tasks on the resulting knowledge base undecidable. To circumvent this, several classes of rules offering various useful properties have been identified. One such class, for instance, contains all sets of rules on which the chase algorithm always terminates, which guarantees the existence of a finite universal model. However, these classes are often abstract rather than concrete: it may be undecidable to check whether a given set of rules belongs to them. Given that the most studied classes of existential rules are designed for reasoning on databases, thus ensuring decidable conjunctive query entailment, we ask: Within a class that supports decidable query entailment, do the usual abstract classes become concrete? We answer in the negative for classes based upon the termination of all classical chase variants and for the bounded treewidth set (BTS) class.