There are two extremes, in my opinion:
- A clear statement by the Pope defining a doctrine that is to be believed by all the faithful everywhere, which is always infallible; and
- A statement as to a particular fact, which may be mistaken (e.g., the Pope may say, “It’s raining outside”, when in fact it is sunny).
A possible case in which a Pope was mistaken as to a particular fact is Pope Adrian I, who seems to have been under the false impression that the canons of the Council of Trullo had been adopted by the Sixth Ecumenical Council, which was not the case.
sourcebooks.fordham.edu/basis/trullo.asp
The real point of discussion is the “in-between zone” - statements not ex cathedra, but not as to a particular fact. There was a party at the First Vatican Council that wanted to adopt the dogma that the Pope is infallible in all his statements (just as there were those who did not believe he was infallible even in ex cathedra statements). The council only defined a dogma with respect to ex cathedra statements - it did not define a dogma with respect to non-ex cathedra statements.
Thus, it may be the case that the Pope is infallible in all his statements. Vatican I did not reject this. It simply affirmed that he is infallible when speaking ex cathedra. The weight of tradition prior to Vatican I strongly supports that the Pope is, at least as a general matter, infallible when speaking about the faith. This type of infallibility seems to have been proclaimed by Pope Agatho I in his letter to the Sixth Ecumenical Council:
newadvent.org/fathers/3813.htm
And Vatican I summed up the tradition concerning infallibility as follows in Canon 6:
"For the Holy Spirit was promised to the successors of Peter not so that they might, by his revelation, make known some new doctrine, but that, by his assistance, they might religiously guard and faithfully expound the revelation or deposit of faith transmitted by the apostles.
Indeed, their apostolic teaching was embraced by all the venerable fathers and reverenced and followed by all the holy orthodox doctors, for they knew very well that
this See of St. Peter always remains unblemished by any error, in accordance with the divine promise of our Lord and Savior to the prince of his disciples: I have prayed for you that your faith may not fail; and when you have turned again, strengthen your brethren."
ewtn.com/library/councils/v1.htm