We consider the predictions of chiral perturbation theory for $SU(3)$ breaking in the axial vector form factor $g_1$ measured in semileptonic hyperon decays. We confirm that if only octet baryon intermediate states are included, the non-analytic corrections are $\sim 100\%$. These corrections are dominated by an $SU(3)$-symmetric wavefunction renormalization, which explains the fact that the ``corrected'' predictions still fit the data well. We argue that the large corrections are nonetheless strong evidence that this chiral expansion is breaking down. Following a recent suggestion of Jenkins and Manohar, we then include contributions from decuplet baryon intermediate states. Unlike these authors, we do not neglect the octet--decuplet mass difference $\Delta$. We find that the effects of $\Delta \ne 0$ significantly change the pattern of corrections: we still find that the decuplet corrections can cancel the large octet contributions in a non-trivial way, but the corrections no longer favor the $SU(6)$ values of the axial couplings. We also argue that $D$ and $F$ axial couplings cannot be reliably extracted from calculations which keep only the non-analytic corrections.