  Let $X$ be a smooth complex projective curve and let $E$ be a vector bundle
on $X$ which is not semistable. We consider a flag bundle $\pi: \text{Fl}(E)
\to X$ parametrizing certain flags of fibers of $E$. The dimensions of the
successive quotients of the flags are determined by the ranks of vector bundles
appearing in the Harder-Narasimhan filtration of $E$. We compute the Seshadri
constants of nef line bundles on $\text{Fl}(E)$.
