We have a finite set $X$, a weight function $w: X\rightarrow \mathbb{Z}^+$, and constants $k\leq c\in\mathbb{N}$. Let the weight $w(S)$ of a set $S\subseteq X$ be the sum of the weights of its elements. Let $A = \{S\subseteq X$ | $k\leq w(S) \leq c \}$.

Form a graph with vertices $A$ and an edge $S-T$ iff $S\cap T\in A$.

Is anything known about the diameters of the connected components of this graph? In particular, is there an upper bound in terms of $|X|$? I know of no case where the diameter exceeds $|X|/2$, for example. I do not claim this is optimal, and any non-obvious bound would be very interesting, especially if it is polynomial in $|X|$.

This question comes from theoretical computer science, but it seems purely combinatorial, and perhaps an expert in extremal combinatorics has some insight into it.