Abstract: The definition of degrees of freedom of a mechanical system originated from the number of independent coordinates to describe the system configuration. The definition turned to be the number of independent variations of generalized coordinates after the studies on non-hololomic constraints in the development of analytic mechanics. The paper points out that the above definition of degrees of freedom has some flaws for the mechanical system with non-holonomic constraints and may impose excessive limits on the system dynamics. The paper, hence, studies the accessible state manifold of a mechanical system with non-holonomic constraints in the state space and shows that the dimensions of the accessible state manifold is equal to the number of minimal unknown variables to describe the system dynamics, governed by a set of ordinary differential equations of the first order, such as the Gibbs-Appell equations together with the relation of generalized velocities and psudo-velocities. Then, the paper defines the degrees of freedom of a mechanical system as a half of the dimensions of the accessible state manifold. Afterwards, the paper demonstrates how to understand the concept of a half degree of freedom of a mechanical system with a single non-holonomic constraint via two case studies, that is, the vibration system having a viscoelastic mounting and the sleigh system moving on an inclined plane, presenting the relation between a half degree of freedom and the two neighboring integer degrees of freedom. Furthermore, the paper gives two examples of mechanical systems, each of which has two non-holonomic constraints and results in the reduction of a single degree of freedom, and addresses the dimensions of tangent and cotangent bundles of the two systems.