Is Infallibility of the Pope or of the Church Biblical?

"Beloved, do not believe every spirit, but test the spirits to see whether they are from God, for many false prophets have gone out into the world."
- I John 4:1

The Church of Rome claims for itself that it is blessed with infallibility, that is, that it is incapable of error. She claims this for Her Church as a whole, and for the Pope in particular. In polemics against the churches of the Reformation, Rome claimed that this infallibility made her naturally superior to the Reformers, who explicitly eschewed any personal infallibility.

However, what do the scriptures say? We see above the statement of the Apostle John: The Christian must test the spirits, never accepting spiritual claims at face value, for there are false prophets in the world. In John 5:39, the same Apostle quotes the Lord's praise for those who search the scriptures for knowledge of Him. A little further in the New Testament in Acts 17:11, the believers of Berea are praised for "examining the Scriptures daily to see if these things were so."

In other words, the scriptures give the exact opposite surety from the Catholic Church: the foundation of our faith isn't from an infallible Pope or infallible Church, but rather from an infallible Bible, for "all Scripture is breathed out by God and profitable for teaching, for reproof, for correction, and for training in righteousness" (II Timothy 3:16).