  We construct an explicit example of a stable bundle on the twistor space
$\mathrm{Tw}(M)$ of a hyperk\"ahler manifold $M$ whose restrictions to all the
fibres of the natural twistor projection $\pi : \mathrm{Tw}(M) \to
\mathbb{CP}^1$ are nonstable. We also describe the relationship between bundles
on $\mathrm{Tw}(M)$ that do not have subsheaves of strictly lower rank and
bundles that stably restrict to the fibres of $\pi$, and announce a result
whose proof will appear in a forthcoming paper.
