  It has recently been shown that fairly strong axiom systems such as
$\mathsf{ACA}_0$ cannot prove that the antichain with three elements is a
better quasi order ($\mathsf{bqo}$). In the present paper, we give a complete
characterization of the finite partial orders that are provably $\mathsf{bqo}$
in such axiom systems. The result will also be extended to infinite orders. As
an application, we derive that a version of the minimal bad array lemma is weak
over $\mathsf{ACA_0}$. In sharp contrast, a recent result shows that the same
version is equivalent to $\Pi^1_2$-comprehension over the stronger base theory
$\mathsf{ATR}_0$.
