Since Paul’s epistles started to be copied the 2nd half of the first century (see Col. 4.16. 2Pet. 3.15-16), transcription has occurred by hand. Gutenberg’s adoption of moveable type in about 1450, and the subsequent printing of bibles shortly after, leaves 1400 years of hand-reproduction of the N.T. texts. Mistakes were inevitable, but, since so many different families of texts were preserved, it is not difficult to ascertain what was originally written.
Textual studies of biblical manuscripts both O.T. and N.T. is very fascinating and involved to say the least. Even a cursory knowledge of transmission can aid understanding of how God preserved His word in providential manner. Needless to say, the process was not cut and dried. However, since God is ultimately directing all events, He knew what He was doing in how He transmitted His word. Here is an article about N.T. manuscripts.Manuscripts 101: What is a Manuscript? — The Center for the Study of New Testament Manuscripts