  Two counterexamples, addressing questions raised in \cite{AD} and \cite{PZ},
are provided. Both counterexamples are related to chaoses. Let $F_n=Y_n+Z_n$.
It may be that $F_n\overset{a.s.}\longrightarrow 0$,
$F_n\overset{L_{2+\delta}}\longrightarrow 0$ and
$E\bigl\{\sup_n\,\abs{F_n}^\delta\bigr\}<\infty$, where $\delta>0$ and $Y_n$
and $Z_n$ belong to chaoses of uniformly bounded degree, and yet $Y_n$ fails to
converge to 0 a.s.
