What would be the configuration space of $n$ particles (each particle position being in $\mathbb{R}^3$), if one insists that it should be such that no two particles can be at the same position?
For two ($n=2$) particles each with position in $\mathbb{R}^1$, I suppose this would be the plane with the line $y=x$ removed from it. This immediately tells us that if the system starts in one region of the space, then it cannot cross $y=x$ to the other. This makes sense since a particle cannot jump over the other, but is special to this case.