Can any counting situation that works for one side of an identity work for the other (combinatorial proof) If I come up with a situation that works for one side of a combinatorial proof, does some interpretation always exist for how the other side counts that same situation?  Or is it possible that one side cannot count a situation that the other side can count, even though they are numerically equivalent?
Take for example the following identity.
$$\frac{(2n)!}{n!(n+1)!}=4\binom{2n-1}{n}-\binom{2n+1}{n}.$$
The left side has a smooth interpretation. 

How to understand this interpretation from the right side?
Take for example the following identity.
$$\frac{(2n)!}{n!(n+1)!}=4\binom{2n-1}{n}-\binom{2n+1}{n}.$$
The left side has a smooth interpretation. 

How to understand this interpretation from the right side?