There's a general folk belief among skeptics in the unreliability of biblical texts, reinforced by the spatter of footnotes at the bottom of modern translations, and excised from the KJV versions that older people grew up with. Now in the OT there's more basis for this. There are areas where there is obvious damage to the Hebrew text. But these are not extensive, and they tend to be concentrated in certain passages and books. In the NT the issues are much smaller.
Of late there has been a lot of fuss about Misquoting Jesus, a book by Bart Ehrman that claims that the New Testament has been redacted (accidentally and on purpose) in the direction of supporting Orthodox doctrine against the competition. I haven't read this book, and I'm not sure how well I could really evaluate it, not being a scriptural scholar nor having ready access to the texts which would need to be cited in order to defend such a thesis.
I have managed to find this discussion of textual reliability in general, with some specific discussion of Ehrman's work (based primarily on a previous book, however). The impression I get from the article is that Ehrman's work is based on comparison of texts of different age; thus his conclusions are generally sound, but also largely irrelevant to readers of modern translations. The reason for this is that the translators, in working from the same text, tend to translate from the older (and presumably unchanged) version. Here and there in the crank-odox world one finds those who are fanatical adherents to the "Byzantine" text rather than the Nestle-Aland older versions favored by most modern translations, and this text (as well as the western Textus Receptus) are presumably subject to the errors/changes Ehrman discusses. As the article I cite comments, most of the small amendations pale against the larger surface message of the text, and anyone reading a modern, non-sectarian translation won't even see most of them. The media seem to be making far more out of this book than is justified.
Rick Laribee cites a dissertation which analyzes the variation in a single passage in detail. It should surprise nobody to learn that the analysis shows that almost all the variation involves easily correctable typos and other obvious errors. Indo-European languages such as Greek allow extensive error correction; the distance between meaningful variations of a text is typically quite large, and therefore requires substantial modification to get from one to the other.
There's really no getting past that the scriptural texts do intend to tell the same story which their authors intended to tell.