This closely related question has an answer that tries to argue in a theological point of view. However, I would like to discuss this topic from a purely grammatical perspective.
πᾶσα γραφὴ θεόπνευστος καὶ ὠφέλιμος πρὸς διδασκαλίαν, πρὸς ἐλεγμόν, πρὸς ἐπανόρθωσιν, πρὸς παιδείαν τὴν ἐν δικαιοσύνῃ, ἵνα ἄρτιος ᾖ ὁ τοῦ θεοῦ ἄνθρωπος, πρὸς πᾶν ἔργον ἀγαθὸν ἐξηρτισμένος.
Can this sentence be interpreted as both inclusive or exclusive?
The exclusive version would be:
All Scripture is divinely inspired and profitable ...
The inclusive version would be:
All divinely inspired Scripture is profitable ...
From Ellicot's[sic] commentary: Although this rendering is grammatically possible, the more strictly accurate translation, and the one adopted by nearly all the oldest and most trustworthy versions (the Syriac and the Vulgate), and by a great many of the principal expositors in all ages (for instance, Origen, Theodoret, Grotius, Luther, Meyer, Ellicott, and Alford), runs as follows: “Every scripture inspired by God is also profitable for doctrine, for reproof,” So this verse itself, given that alternate translation may answer your question.
Case for the exclusive interpretation
The "καὶ" connects two adjectives, both in the nominative feminine singular: "θεόπνευστος" and "ὠφέλιμος". Then, it is clear that both adjectives are describing the noun "πᾶσα γραφὴ" simultaneously.
The Vulgate translation reads:
omnis scriptura divinitus inspirata et utilis ad docendum ad arguendum ad corrigendum ad erudiendum in iustitia ut perfectus sit homo Dei ad omne opus bonum instructus
which contradicts the quote above from Ellicott's commentary, as this is clearly the exclusive interpretation.
Case for the inclusive interpretation
This requires interpreting "πᾶσα γραφὴ θεόπνευστος" as the whole subject and "καὶ" as an adverb meaning "also".
In Matthew 7:17:
οὕτως πᾶν δένδρον ἀγαθὸν καρποὺς καλοὺς ποιεῖ, τὸ δὲ σαπρὸν δένδρον καρποὺς πονηροὺς ποιεῖ·
Here, "πᾶν δένδρον ἀγαθὸν" is in the construction of "every + [noun] + [good]" as a single noun phrase, just like the first three words of our verse in question.
However, "καὶ" is not an adverb, but a conjunction, which seems to render this interpretation grammatically impossible.