Formally this is:
$\exists S [\forall \mathcal M (\mathcal (\mathcal M\models T) \to (\mathcal M \models S)) \wedge \neg (T \vdash S)]$
In English: there is a sentence in the language of an effectively generated theory $T$ that is satisfied in every model of $T$, and yet not provable in $T$.
Can this occur in some effective logical systems other than first order logic? What are those systems?
[EDIT]In first order logic systems this cannot happen because of Godel's completeness theorem for first order logic, which implies that satisfiability corresponds to syntactical provability, so if a sentence is satisfiable in every model of a first order theory, then it must be a provable in that theory.