Coordination is often a relevant thing with regards to identifying the syntactic structures underlying a language. A question like 'are indirect and direct objects the same thing' can in some languages be decided to be negative because the two cannot be coordinated:
*I gave him and a gift
We also find that we can't coordinate subjects and objects in English:
*I and him hit
However, we can also imagine semantic restrictions on objects, e.g. forbidding something like
he kicked the wall and the man
due to the difference in animacy (or whatever) being too large. Sometimes, we find that different cases can coordinate, e.g. with locatives or certain other obliques:
he is in jail and out of luck
However, these usually are of similar phrase-types, e.g. adverbials or complements or whatnot.
We could go and do weird things with this though. What if the singular and the plural version of, say, the object case, could not be coordinated?
*I saw her and them
*I saw the teacher and the pupils
This would force some periphrasis, or alternatively there'd be morphological cheats - maybe possessive markers hide the plural accusative marker (like in Finnish), and permits it again, so you could say 'I saw the teacher and his pupil(s)'. Or maybe a 'fake pluralizing' strategy emerges, where singulars can have a formally plural but semantically empty pluralization going? A depluralizer of sorts: maybe the numeral 'one' with plural congruence on it?
I saw the one-s teacher-s and the pupil-s