Can logical operators be used with Cases?

Generally I would use Condition:

Cases[Range[-75, 100], x_ /; Positive[x] && PrimeQ[x]]

For the specific case of And you can string PatternTest if you control grouping:

Cases[Range[-75, 100], (_?Positive)?PrimeQ]

{2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47,
 53, 59, 61, 67, 71, 73, 79, 83, 89, 97}

Cases[Range[-75, 100], _?(Positive[#] && PrimeQ[#] &)]

(* or *)

Cases[Range[-75, 100], _?(Apply[And, Composition[Positive[#], PrimeQ[#]]] &)]
(* which is the same as the one below*)

Cases[Range[-75, 100], _?(Apply[And, Positive[#1]@*PrimeQ[#1]] &)]

(* or even more compactly *)
Cases[Range[-75, 100], _?(And @@ Positive[#1]@*PrimeQ[#1] &)]

a different way with Cases:

(Cases[#, _?Positive] ⋂ Cases[#, _?PrimeQ]) &@Range[-75, 100]

You can use operator forms with Select:

Select[
    Range[-75,100],
    Through @* And[Positive, PrimeQ]
]

{2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97}