For any three distinct vertices,

$$c(u,w)\ge \min\{c(u,v),c(v,w)\}.$$

Indeed, removing edges to disconnect $u$ from $w$ must also disconnect either $u$ from $v$ or $v$ from $w$. Otherwise, the remaining paths through $v$ would still connect $u$ to $w$.

For each positive integer $t$, put two distinct vertices in the same class exactly when $c(u,v)\ge t$. The inequality makes this relation transitive, so it defines a partition.

As $t$ increases, classes can split but cannot merge. Consequently, any two classes appearing at any thresholds are either disjoint or one contains the other.

Let $M=\max_{u\ne v}c(u,v)$. At threshold $1$, all vertices belong to one class. At threshold $M+1$, every class is a singleton.