Can you use the effects of several berries at the same time?

Solution 1:

Nope, only one berry can be used at a time during a Pokémon encounter, as mentioned in this Reddit thread:

You can only use 1 berry at a time on a pokemon. So no, you can't use a razz + a nanab on a pokemon. The active berry is used as an icon on the pokemon. Once the icon disappear, on a failed catch attempt, you can use an another berry.

For a visual representation of what happens:

Solution 2:

No. If you use a berry and its effect is in place, the other berries in the berry carousel can't be selected until the effect has been used (i.e. the next capture attempt).