  We prove a Kadec-Pelczy\'nski dichotomy-type theorem for bounded sequences in
the predual of a JBW*-algebra, showing that for each bounded sequence
$(\phi_n)$ in the predual of a JBW$^*$-algebra $M$, there exist a subsequence
$(\phi_{\tau(n)})$, and a sequence of mutually orthogonal projections $(p_n)$
in $M$ such that: [$(a)$] the set ${\phi_{\tau(n)} - \phi_{\tau(n)} P_{2}
(p_n): n\in \mathbb{N}}$ is relatively weakly compact,
$\phi_{\tau(n)}=\xi_n+\psi_n$, with $\xi_n := \phi_{\tau(n)} - \phi_{\tau(n)}
P_{2} (p_n)$, and $\psi_n := \phi_{\tau(n)} P_{2} (p_n),$ {\rm(}$\xi_n Q(p_n)=
0$ and $\psi_n Q(p_n)^2 = \psi_n${\rm)}, for every $n$.
