The driven-cavity problem, a renowned bench-mark problem of computational, incompressible fluid dynamics, is physically unrealistic insofar as the inherent boundary singularities (where the moving lid meets the stationary walls) imply the necessity of an infinite force to drive the flow: this follows from G.I. Taylor's analysis of the so-called scaper problem. Using a boundary integral equation (BIE) formulation employing a suitable Green's function, we investigate herein, in the Strokes approximation, the effect of introducing small “leaks” to replace the singularities, thus rendering the problem physically realizable, The BIE approach used here incorporates functional forms of both the asymptotic far-field and singular near-field solution behaviours, in order to improve the accuracy of the numerical solution. Surprisingly, we find that the introduction of the leaks effects notably the global flow field a distance of the order of 100 leak widths away from the leaks. However, we observe that, as the leak width tends to zero, there is exellent agreement between our results and Taylor's thus justifying the use of the seemingly unrealizble boundary conditions in the driven-cavity problem. We also discover that the far-field, asymptotic, closed-form solution mentioned above is a remarkably accurate representation of the flow even in the near-field. Several streamline plots, over a range of spatial scales, are presented.