"question the possibility" is correct and can be used in written English.
You can use it to introduce doubt about a hypothetical situation or outcome. For example, "I'm not sure whether I'll have enough time to finish the project on time. I have to question the possibility."
Some commentators would question the possibility of a direct read-across from local councils to the epic scale of central government.
"It therefore calls into question the possibility that some or all of the decisions he made when passing on disability claims are suspect, and perhaps invalid".
What is lost is the back-and-forth, the follow-up question, the possibility that something unrehearsed will make it into the article.
Postmodern philosophers and philosophers of education challenge basic aspects of traditional philosophical theorizing by calling into question the possibility of objectivity, the neutrality of reason, the stability of meaning, and the distinction between truth and power.
In a floor speech, Mrs. Boxer again rejected the Republican criticism, although her amendment suggested that there might have been some reason to question the possibility of future bailouts.
In a world of disconcerting change, when large and complex forces threaten familiar and comfortable guideposts, the natural impulse is to grab hold of the tree trunk that seems to have the deepest roots and hold on for dear life and never question the possibility that it's not going to be the source of your salvation.
We can't computerize the delicate chemical balance of our neurology that leads us to pursue, and question, the possibilities machines can unlock as "allies" in the 21st century.
Beyond the references to Bayou culture, urban and rural decay, the installation and photographs seem to question the possibilities and contradictions of emotion and life.
Image Atlas highlights cultural differences and similarities, and questions the possibility of a universal visual language.
The new theories questioned the possibility of originality in a world controlled by systems of power and polluted with an overflow of images.
Third, a line of thought pursued by Michel Foucault and later postmodern philosophers questioned the possibility of objectively valid political values and genuinely neutral political institutions.
