The basic constraint of the DSN is that the antennae are highly directional, so you can only receive data from one place at a time. The directionality is a direct consequence of their sensitivity - achieved by using huge parabolic antennae.

If different missions are in different places in space, then you can use different antennae to communicate simultaneously, but if they are too close in space, you can't because there aren't enough antennae. There are no close-spaced duplicates, they are spread widely round the globe to give 24/7 coverage of key targets.

There is a backup in smaller antennae sizes, but deep space missions rely on big antennae with good signal:noise ratio to get high data rates.

For instance, the Mars rovers can only communicate one at a time direct with Earth (though in that case we add the option to relay via MGS, Odyssey, and even MEx).