To answer this question we need to begin with the understanding that as Christians our trust is grounded in the Lord Jesus Christ. While we have many reasons to accept and believe that the Bible is accurate, we can never fully prove it to be so because we do not have the original manuscripts. But what we do have is the promise of God to preserve His Word (Matthew 4:4). This is perhaps an even greater blessing. We think that it would be great to have the original manuscripts so we could prove to the world that the Bible is true. However, if that were so, we would be trusting in the proof of those original manuscripts. Instead, we are trusting our God. Our faith is built on trust in Jesus Christ. He is the foundation from which all our understanding and reasoning comes. Therefore, we walk in faith knowing that He promised to preserve His Word and that His Word is true. "Sanctify them through thy truth: thy word is truth" (John 17:17).
Beginning with that understanding that our trust is first in the God of the Word, let's address how we know that we can also trust the Word of God. The Bible has been translated into many languages and into multiple versions in differing languages. The answer to whether every translation of the Bible is 100% accurate, would certainly be, no. Some Bible translations are definitely more accurately translated than others.
The question asked is likely addressing the multitude of English language Bible available today, most of which are reliable and accurate translations, but some of which are not. In determining which translations are reliable and accurate, one must consider the methods of translation and the manuscripts from which a translation derive.
In the English language, we have a number of time-tested Bible translations that have been proven to be accurate and trustworthy. The King James Version and the New King James come from a line of manuscripts that originated in Antioch, Syria. Throughout the centuries the KJV has been held to be an accurate translation. It was translated from more than 5,200 manuscripts available in 1611. The New King James is very similar to the King James but is written in modern language, removing the Elizabethan language of 1611. Since its publication in 1982, it has been considered by most to be an accurate translation. The New American Standard (NASB, 1971, updated in 1995) is translated from manuscripts that originated in Alexandria, Egypt and is also considered to be a very accurate translation. It is fair to say that these three Bibles are the preserved Word of God and are accurate in their translation. Each of these Bibles is a word-for-word, or formal equivalence, translation. There are also other Bibles that are thought-for-thought, or dynamic equivalence, translations. There will be more variation from the original manuscripts in such Bibles.
It is correct that the Bible has been translated into English many times. However, unlike the game of "telephone" where a message is passed from person to person and can conclude with a very distorted message, in Bible translation the translators begin with the oldest manuscripts available to them. It would be like the end person in the "telephone" game going to the first person and being given the message directly. It would be extremely accurate as it would have lost nothing through the progressive passing from one person to another.
All other English Bibles derive from one of the two manuscript lines — Antioch or Alexandria. There are far too many to discuss their merits and judge which are considered to be accurate and reliable and those that take too much liberty in translation and deviate from the historical manuscripts.