How much ambiguity for argument roles is possible without sacrificing comprehensibility?

For the purposes of this question, I’m specifically interested in marking theta-roles of verb arguments and possessive constructs, i.e. the things prototypically marked by case in languages that have extensive case systems. Many languages use some combination of word order, case marking, argument indexing on the verb, and explicit adpositions to disambiguate/identify arguments within a clause.

Some languages consistently do not mark certain distinctions like subject vs direct object or between different kinds of objects. Yagua makes no distinction between the non-subject core arguments of ditransitive verbs.

I’m wondering how minimal or ambiguous the morphology/syntax relevant to argument roles in a language can be while still being comprehensible.