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Given a holomorphic vector bundle $E$ on the twistor space $\mathrm{Tw}(M)$of a simple hyperk\"ahler manifold $M$, we view it as a family of bundles$\left\{E_I\right\}$ on the fibres $\pi^{-1}(I)$ of the twistor projection $\pi: \mathrm{Tw}(M) \to \mathbb{CP}^1$, and study the relationship betweenstability of $E$ and its fibrewise stability. We verify that the argument ofTeleman establishing the Zariski openness of stability and semi-stability infamilies of bundles applies in the case of the family $\left\{E_I\right\}$. Weprove a partial converse to a result of Kaledin and Verbitsky, showing that anirreducible bundle $E$ on $\mathrm{Tw}(M)$ is generically fibrewise stable ifthe rank of $E$ is 2 or 3, or at least one element of the family$\left\{E_I\right\}$ is a simple bundle, in the sense that $\mathrm{Hom}(E_I,E_I) = \mathbb{C}$.

Families of stable bundles on the fibres of the hyperk\"ahler twistor projection