Friction significantly influences the mechanical system dynamics, especially when self-locking property is observed. The Coulomb model is frequently adopted to represent friction in multibody analysis and simulation. It can be shown that in some extreme cases of joint friction modeling, problems with solution uniqueness and existence are encountered, even when only bilateral constraints and kinetic regime of friction are considered. These problems are studied in detail in the paper. To approach the investigated subject, a wedge mechanism, viewed as a simplified model of a speed reducer, is studied. Two different mathematical models of joint friction are used, both based on the Coulomb friction law. The first version of the model is purely rigid, i.e., no deflections of the mechanism bodies are allowed. Constraints are imposed to maintain the ratio between input and output velocity. The second version of the model allows deflection of the frictional contact surface, and forces proportional to this deflection are applied to contacting bodies (no constraints to maintain the input–output velocity ratio). Using the rigid body model, it is shown that when friction is above the self-locking limit, paradoxical situations may be observed when kinetic friction is investigated. For some sets of parameters of the mechanism (gearing ratio, friction coefficient, and inertial parameters), two distinct solutions of normal and friction forces can be found. Moreover, for some combinations of external loads, a solution that satisfies equations of motion, constraints, and the Coulomb friction law does not exist. Furthermore, for appropriately chosen loads and parameters of the mechanism, infinitely many feasible sets of normal and friction forces can be found. Investigation of the flexible body model reveals that in nonparadoxical situations the obtained results are closely similar to those predicted by the rigid body model. In previous paradoxical situations, no multiple solutions are found; however, problems with stability of solutions emerge. It is shown that in regions for which the paradoxes were observed only unstable solutions are available. The origins of paradoxical behavior are identified and discussed. The key factors determining the model performance are pointed out. Examples of all indicated problematic situations are provided and analyzed. Finally, the investigated problems are commented from more general perspectives of multibody system dynamics.