  We analyze the origin of the structures observed in the reactions $e^+e^-\to
3(\pi^+\pi^-)$, $2(\pi^+\pi^-\pi^0)$, $\omega\pi^+\pi^-\pi^0$, and $e^+e^-\to
2(\pi^+\pi^-)\pi^0$ around the antiproton-proton ($\bar pp$) threshold. We
calculate the contribution of the two-step process $e^+e^-\to \bar NN \to$
multipions to the total reaction amplitude. The amplitude for $e^+e^-\to \bar
NN$ is constrained from near-threshold data on the $e^+e^-\to \bar pp$ cross
section and the one for $\bar NN \to$ multipions can be likewise fixed from
available experimental information, for all those $5 \pi$ and $6\pi$ states.
The resulting amplitude for $e^+e^-\to$ multipions turns out to be large enough
to play a role for the considered $e^+e^-$ annihilation channels and, in three
of the four reactions, even allows us to reproduce the data quantitatively near
the $\bar NN$ threshold. The structures seen in the experiments emerges then as
a threshold effect due to the opening of the $\bar NN$ channel.
