It is well known that the permeability of a set of joints can significantly vary in response to in situ stress conditions and pressure of the flowing fluid. Frequently, joint sets are closely spaced, and although joint mechanical interaction could significantly affect their aperture, the interaction is usually ignored in the fluid flow models. It is rather obvious that this approach corresponds to the upper bound for flow rate and rock permeability. By taking into account the interaction between the joints, we show that modeling a joint set by an infinite array provides the lower bound. The difference between these bounds, however, can be rather large, so they may not always be used with the sufficient accuracy. From the conceptual standpoint, it is often tempting to model a set with a finite number of joints by an infinite array. The results obtained in this work clearly demonstrate that such a model may result in a significant underestimation (by orders of magnitude) of both the permeability and flow rate. Similarly, the assumption of noninteracting joints may significantly overestimate (also by orders of magnitude) the stress-dependent permeability and flow rate compared to those computed more accurately when accounting for joint interaction. Because the internal pressure can, in fact, close the pressurized joints while two edge joints (end-members) in the set remain widely open (since they are not suppressed from one side by the adjacent joints), unless the number of joints in the set is exceedingly large (typically, >103), the fluid flow through the joint set becomes highly heterogeneous, focusing in the edge joints. As a result, the permeability/flow rate dependence on the joint spacing is not monotonic but has a maximum and a minimum. The derived closed-form expression for flow rate/permeability ratio is asymptotically accurate and allows computations for rather arbitrary joint sets.