Just a quick lazy answer.

By the interlacing property of Schur complements, for a vector $v$ with unit norm one has $\lambda_{\min}(X) \leq \lambda_{\min}(A-B C^{-1}B^T) \leq v^TAv - v^TB C^{-1} B^T v \leq \lambda_{\max}(A) - \frac1{\lambda_{\max}(C)}\|B^T v\|^2$, which gives the bound
$$
(\sigma_{\max}(B))^2 \leq (\lambda_{\max}(A) - \lambda_{\min}(X))\lambda_{\max}(C).
$$
I hope I'm not mixing up any min/max here, but in any case you get the idea of the reasoning.

It should be possible to find examples in which all equalities hold. Probably a similar inequality can be set up for the other direction, too ($\sigma_{\max}(B) \geq \sigma_{\min}(B) \geq \dots$).