"The plates are by the cups."
Translation:A tányérok a csészék mellett vannak.
I would consquently translate -nál as "by/at" and mellett as "next to / beside". Which is not met here, so I think this should be changed, too.
Mellett has a directional sense. I think the meaning can be captured well with "left or right from". Houses in a street tend to stand next to each other, so for neighbouring houses you'd use mellett, for instance when you say you live next to the library (a könyvtár mellett).
The suffix -nál doesn't have that direction, it's just "closeby", like someone at the door (az ajtónál), or you can stand by the river (a folyónál). It also suggests some kind of interaction if the subject is a person.
In many cases they're interchangable, you can stand next to a tree or by a tree. Doesn't make a difference here.
In this special case using mellett would imply to me that you're being told where the plates are to be found in the cupboard: neatly stacked up next to the cups. If you said a csészéknél, it sounds to me like a description of a table after a party, a bit chaotic, but the plates can be found in the general area where the cups are scattered, too.