If the frictional force between two surfaces is insufficient to prevent motion despite an increase in the normal force, what is the underlying cause of this failure in maintaining equilibrium?

When you raise the normal force, the maximum possible static friction grows proportionally to μs N, but only up to the product of the coefficient of static friction and the normal force. If the applied horizontal force is larger than this maximum, the static friction cannot keep the objects still. The failure is therefore not due to a lack of contact but to the fact that the coefficient of static friction is too small to supply the needed resisting force. In other words, the required friction exceeds μs N, so equilibrium is lost. For example, pushing a box on a rough floor with a force of 30 N when μs = 0.