As @Yellow Sky mentioned, based on your assumption that case exists in English, let us call it Abstract Case, then the following case pronominal distribution can be found:
Nominative abstract case (or subjective case): exclusive of finite clauses (tensed clauses); e.g.,
(1) I tabled the data.
In (1), I is nominative in the subject position of a finite clause (past).
Accusative abstract case in subject position of infinitival clauses (a clause without tense); e.g.,
(2) I want him to leave now.
In (2) him is in the subject position of an infinitival clause. Infinitival clauses lack capacity to assign case. So, their subjects are case marked from an outside lexical element, in this case the verb want.
Accusative abstract case in subject position of prepositional complementizers; e.g.,
(3) For him to leave would be terrible.
Again this last example is similar to (2) because the pronoun is found in subject position of an infinitival clause.
Accusative abstract case in object position of preposition; e.g.,;
(4) She moved towards him.
Accusative abstract case in object position of a verb; e.g.,
(5) She kissed him.
Accusative abstract case in object position of ditransitive (double object construction); e.g.,
(6) She sent him a letter (orig. She sent a letter to him).
I think these are the only possibilities for pronouns. A genitive case is exclusive of nouns.