An incompressible steady-state formulation of the Lattice Boltzmann Method is applied to laminar flows for a varying range of Reynolds numbers, extending form 50 to 2000. As test cases, the channel and the lid driven cavity flow problems are considered. The effect of the model Mach number on the accuracy is also analyzed by performing computations for different Mach numbers varying within the range 0.1 - 0.4, comparing the results with each other and with the results obtained by a finite-volume discretization of the incompressible Navier-Stokes equations. For both test cases, it is observed that the implied Mach number by the method does not effect the results within the above-mentioned ranges. An important purpose of the study has been to explore the stability limits of the method. Within this context, it is observed that the largest allowable collision frequency decreases with increasing Reynolds and Mach numbers. It is additionally observed that these dependencies are stronger, and the limiting collision frequencies are lower for the channel flow, compared to the lid driven cavity flow.