Could parts of the bible have been lost over time?

Yes, there are many sacred writings that have existed that do not show up in today's Bible. Here is a list of references in the Bible to these other writings.

In addition, many writings in the Bible today are not as they were written originally, for a variety of reasons.

God is concerned about this. What we have today in the Bible is NOT all that He wants us to have in this day. To solve this problem He has called prophets in our day who have revealed additional scripture that has the purpose of strengthening the Bible's testimony of Christ and helping bring us to Him. The most central of these is "The Book of Mormon: Another Testament of Jesus Christ".

As an example, please see this chapter of the Book of Mormon that was written in about 550 BC that discusses the need for additional scripture to strengthen the Bible as it will exist in our days.

The Bible should be understood as a collection of writings that have various theology relevant pieces of historical writings in them. We call it a book, but this is a misnomer. It is a collection of various writings relevant to the Christian faith.

The old covenant explains the Jewish roots that Jesus came from. It also provides the various parts of prophecy that Jesus came to fulfil. The new covenant explains how the wandering Jew came to fulfil all those ancient prophecy.

The letter to the Corinthians is nothing else than a letter. Corinth is a real place. You could probably still visit the place were the church in Corinth stood. All these Biblical stories refer to figures in history that were very much real. The Bible is very much a historical document.

So then when you realise that this 'book' is a collection of the most relevant and reliable historical documents regarding the teachings of Jesus of Nazareth it should be no surprise that there were less reliable flights of fancy in circulation in and around Jesus time. Just like today anyone could write something about Jesus.

You can still read these writings. They read like ancient fan fiction to me. No doctrine important issues are contained with them. They are all not in the Bible just because they contain nothing relevant, because they are from less reliable sources or they contradict a generally accepted thing believed by Jesus by the people who knew him.

Overall it's not possible to prove something does not exist, so there is really no answer to that question.

On the other hand it means that it is possible. There are also known biblical "apocrypha" books, which look like they were part of the Bible but in the end were often deemed either fake or just not logically consistent with the rest of the Bible. The Bible consists only from the materials that were deemed internally consistent and authentic. Those apocryphal books were not destroyed nor hidden and the criteria were clear, so you can just look them up and decide for yourself if you want to live by them.

Also when it comes to the "completeness" of the Holy Word. There is a line in "Matthew 7:16 By their fruit you will recognize them. Do people pick grapes from thornbushes, or figs from thistles?". What is there in the Bible was enough to produce countless saints and solve many spiritual issues, so I think what we managed to preserve is at least good enough guidance for Christians.

Also my personal opinion: I think we should not fall into extremes like "all or nothing". We should do the best with what we have. Assuming the God exists, He will help us to fill in the blanks along the way if there are any, but we need to act, not just expect that to happen. If we assume the Bible is incomplete and we shouldn't act on the incomplete information that would just be paralyzing and would lead to no results. Also, even if the information is incomplete (there is no way to prove nor disprove this), compare this to science. Science also acts on incomplete information all the time and produces results, so that doesn't really affect much. Personally I believe the Bible is complete as is, as I personally never ran into any unexplainable dead-ends while reading it, but it's still just an opinion.

The Bible tells us that they were lost - then found! In the days of Josiah, they found the book of the Law which was gathering dust in a storeroom, brought it to the king and it caused quite a stir.

First, consider the New Testament. Jesus famously said, "Heaven and earth will pass away, but my words will never pass away." (Matthew 24:35) Thus we can be sure that we are not missing any gospels.

Now for the Old Testament.

This article (which advocates documentary theories which I dispute, but makes good points) offers information relevant to the discussion:

https://www.haaretz.com/jewish/2019-05-27/ty-article/.premium/who-was-behind-the-canonization-of-the-hebrew-bible/0000017f-e328-df7c-a5ff-e37a66090000

Jesus spoke of the Law, the Prophets and the Psalms (shorthand for the Writings, of which Psalms was longest), indicating his acceptance of that division of the Hebrew Scriptures as comprehensive.

44 He said to them, “This is what I told you while I was still with you: Everything must be fulfilled that is written about me in the Law of Moses, the Prophets and the Psalms.” (Luke 24:44)

The cited article above speaks of the schism between the Samaritans and the Jews shortly after the time of Ezra. The Samaritans only accept the Torah (five books of Moses) and not the prophets or writings. This schism and the consequent closing of the Samaritan canon tells us that the Torah part is complete.

Then just before Alexander the Great's invasion, the section devoted to the prophets was settled. Zechariah's prophecy foretold a time when there would be no more prophets:

“On that day every prophet will be ashamed of their prophetic vision. They will not put on a prophet’s garment of hair in order to deceive” (Zechariah 13:4)

Thus the Jews rejected all prophets following this era. None of the accepted prophets use Greek borrow words or show Hellenistic influences, so we have evidence that no late prophets were added accidentally. Thus we have a second stage in the establishment of the Old Testament canon. Since Jesus is called the word of God, quotes the Old testament Prophets, and makes no statement about missing or surplus prophets, we can be sure that the list of prophets is complete.

The third stage, the formation of the writings, took longer. The Jews argued about Song of Songs and Ecclesiastes the longest. In the 2nd century AD, Josephus stated that there were 22 books in the Old testament. Between the 2nd and 3rd centuries AD those last two books were finally accepted.

Given that Jesus recognized three categories of Scripture and he is God, there are no missing categories. Given the formational process, we can be sure that we are not missing parts of the Torah (books of Moses) or any prophets, or Jesus would have said so. Since the writings were not settled until after Jesus, that is the only area for debate.

In my book Peace, like Solomon Never Knew, I uncover two structural and thematic patterns that run through these seven Books: Psalms, Job, Proverbs, Song of Songs, Ecclesiastes, Matthew and Revelation. By that evidence and other similarities, those seven books can be shown to be the Seven Pillars of Wisdom spoken of in Proverbs. They are thus complete and interlocking, meaning no possibility of any extra wisdom books being absent and certainly proving the suitability of Ecclesiastes and Song of Songs being part of the canon, as well as Revelation, which was challenged by the Council of Laodicea and others.

As for books subsequent to Revelation, it is interesting that many of the Seven pillar books have warnings against adding to God's words, including Ecclesiastes and Revelation. We no longer have apostles who knew Christ, so we should not expect that we are missing any recent Bible additions.

Tackling Paul's letters and the rest of the New Testament is more than I can handle now. You should consult other articles on theories of Bible canon formation.