Locally accessible dimensions of configuration space

I am reading a book called “Structure and Interpretation of Classical Mechanics”
by MIT Press.While discussing configuration space and degrees of freedom,the authors remark the following:

Strictly speaking, the dimension of the configuration space and the number of degrees of freedom are not the same. The number of degrees of freedom is the dimension of the space of configurations that are “locally accessible.” For systems with integrable constraints the two are the same. For systems with non-integrable constraints the configuration dimension can be larger than the number of degrees of freedom.

I am aware of the distinction between holonomic and non-holonomic consraints.But I can’t relate what the authors mean by locally accessible dimensions.

Can someone shed light on this matter ?A dynamical system which explains the idea would be appreciated.