Why are the books of the Protestant Bible in that order? Is there any sense behind it?
Well, yes. But not as much as you might think.
Firstly, it is worth remembering that different groups of Jews and Christians put their Old Testament books in different orders. The New Testament is pretty consistent across Christian traditions. No group claims that the order of Biblical books (called the “canonical” order) is divinely inspired of God. Additionally, Catholics and Eastern Orthodox Christians include books that Protestants do not.
However, the Protestant Bible was arranged with some thought. The Old Testament is arranged chronologically (in the order the events happened or the books were written) within categories. There are 5 categories: Mosaic books, history, writings, major prophets, and minor prophets.
It begins the same in almost every tradition. The 5 books of Moses have been placed in the order—Genesis, Exodus, Leviticus, Numbers, and Deuteronomy—since the beginning. Since these books end at the entrance of the Promised Land, it seemed logical to include the books about the conquering of the Promised Land next—Joshua and Judges. Ruth occurs during the period of the Judges, so it is next. Following the taking of the promised land, we have the last judge, Samuel, and the first kings. Thus, 1-2 Samuel, 1-2 Kings. Then the story of the southern kingdom is re-recorded after their return from exile in 1-2 Chronicles. Ezra and Nehemiah record the actual history of the return from exile. Esther occurs simultaneous with Ezra but in a different location, so it is next. At this point, the Old Testament history books come to a conclusion
Poetic writings come next with Job, Psalms, Proverbs, Ecclesiastes, and Song of Solomon. Job likely occurred during Genesis. The Psalms were mostly written by David, and the last three books are attributed to his son Solomon. Thus, these are arranged chronologically.
Finally, the longer prophetic books are followed by the shorter prophetic books. Isaiah warns Judah as does Jeremiah later. Jeremiah laments about the fall of Judah in Lamentations. Ezekiel and Daniel are written during the Babylonian Exile. The minor prophets begin by warning Israel, then Judah, then talking about the exile, and concluding with its return to Judah and the rebuilding of the Temple.
The New Testament is less chronological. Matthew starts by talking about the birth of Jesus and his connection to the Old Testament, thus it is placed as a connector between the two Testaments. Mark, Luke, and John are placed in the order they were likely written, though they speak about the same time period. Acts, though written by Luke, is placed after John to keep events in order.
Then the letters begin. Paul's letters come first, arranged basically longest to shortest from Romans to Philemon. Hebrews is of unknown authorship, but just in case it was by Paul, this letter is placed next. The order of the books of James, 1-2 Peter, 1-3 John, and Jude makes no sense to me. Chronologically, topically, and categorically it makes way more sense to put 1-3 John next to Revelation and have Jude next to 2 Peter. Of course Revelation was the last New Testament book written and talks about the future.
So you see, it isn't entirely cut and dry. Of course there are chronological Bibles out there that make an attempt to straighten this all out. However, You do run into challenges like Job. Its events occurred either pre-flood or in the patriarchal time period (Genesis), but it probably wasn't written until the period of wisdom literature that includes Proverbs and Ecclesiastes.
The order of the books isn't extremely significant. Just pick one and read it!