Can the Bible be added to?
The Bible is not one book at all, but many books, and so anytime any one book speaks about "Scripture", we must remember the scope of the statement. When St. John in his Revelation says that no one shall add to or change the book (Revelation 22,18) there is no reason to think he is talking about anything besides the book he is writing. This is especially because Revelation was not part of the Bible until long after it was written. Similary, when Ezra mentions that scripture is fixed, he is most likely talking about the Torah (just the first five books of the Bible, i.e. the Pentateuch), because the rest of the books of the Old Testament were not considered part of the Canon (how could they be, they weren't all written yet!). The same applies to every statement made by St. Paul regarding scripture: he wasn't talking about the New Testament. The New Testament didn't formally exist! If Paul really meant that nothing could be added to scripture, then why were his letters added?
Conclusion: I see no reason why not.
Why is the Bible the way it is today?
There are also many books that could have been added to both the Old Testament (such as the Dead Sea Scrolls) and the New Testament (the Gospel of Thomas, the book of Mormon I guess), but weren't. So how do we know what books belong and what do not? According to the Catechism of the Catholic Church:
120 It was by the apostolic Tradition that the Church discerned which writings are to be included in the list of the sacred books. This complete list is called the canon of Scripture. It includes 46 books for the Old Testament (45 if we count Jeremiah and Lamentations as one) and 27 for the New.
The Old Testament: Genesis, Exodus, Leviticus, Numbers, Deutoronomy, Joshua, Judges, Ruth, 1 and 2 Samuel, 1 and 2 Kings, 1 and 2 Chronicles, Ezra and Nehemiah, Tobit, Judith, Esther, 1 and 2 Maccabees, Job, Psalms, Proverbs, Ecclesiastes, the Song of Songs, Wisdom, Sirach (Ecclesiasticus), Isaiah, Jeremiah, Lamentations, Baruch, Ezekiel, Daniel, Hosea, Joel, Amos, Obadiah, Jonah, Micah, Nahum, Habakkuk, Zephaniah, Haggai, Zechariah, and Malachi.
The New Testament: Matthew, Mark, Luke and John, the Acts of the Apostles, the Letters of St. Paul to the Romans, 1 and 2 Corinthians, Galatians, Ephesians, Philippians, Colossians, 1 and 2 Thessalonians, 1 and 2 Timothy, Titus, Philemon, the Letter to the Hebrews, the Letter of James, 1 and 2 Peter, 1, 2, and 3 John, and Jude, and Revelation (the Apocalypse).
The
Church set the canon unofficially at the
Council of Hippo and the
Council of Carthage, and officially at the
Council of Trent. Although all the books in the Canon are divinely inspired, and inerrant in matters of faith and morals,
not necessarily every inspired book is in the Canon. Presumably, the Church could change the official list of books, but it is unlikely to ever do so.
Those who do not accept the authority of The Catholic Church (e.g. Orthodox Churches, the Church of Jesus Christ of Latter-Day Saints) can set their own canons; I see no scriptural reason against it (other than the biased one that the Catholic Church is the only true guardian of Christian Tradition, and since she was the Church established by Christ Himself, she alone can say what should and what should not be included in the Canon of Scripture). Most Protestants, who believe they don't need Church authority to interpret the Bible, may as well add and delete any books they see fit. (I wonder why more Protestants don't change the fixed set of 27 books that exist in the New Testament precisely because Councils of the Catholic Church established them).
The Protestants traditionally have the same New Testament, but their Old Testament excludes Tobit, Judith, 1 and 2 Maccabees, Wisdom, Sirach (Ecclesiasticus), parts of Daniel, parts of Esther, etc. The general idea is that these works are not accepted by most Jews; many of them only existed in Greek or Aramaic form (as opposed to Hebrew) for centuries. The Protestant Old Testmament contains 39 books.
for more see the Nodes: Does the Bible allow for Additional Mormon Scriptures?, King James Bible, and Apocrypha.