Hasn't the Bible been translated and copied so many times that what we have today cannot be trusted?

Of course it's true that the Bible has been translated into a vast number of languages. In fact, the Bible is the most translated book in the world, and in many cases a language was first put into writing by Christian missionaries for the express purpose of making the Bible available to a specific people group. And it's also true that the Bible was repeatedly copied by hand for many hundreds of years (until the advent of the printing press, invented in Europe in the 15th century). In both cases, mistakes were made.

Far from making the Bible unreliable, however, the vast number of translations and thousands of manuscripts still in existence help to ensure the accuracy and reliability of our modern Bible. How so? Through the science of textual criticism, which can be defined as: "Comparison and evaluation of the different readings of the manuscripts of the Bible in order to construct the history of the text through its various stages and to establish, as far as possible, the original." (Nelson's New Christian Dictionary)

In the case of the New Testament, today there exists nearly 5,700 Greek manuscripts, some fragments of which can be dated to within only a few years of Jesus' death and resurrection. This number has been said to be embarrassingly high when compared with other ancient writings. While no two manuscripts are exactly alike, modern Bible scholars have arrived at a text that is very nearly 100% accurate. What's more, any questions or uncertainties that remain do not impact any major Bible doctrine. (A General Introduction to the Bible)