The NLT, like most modern versions, is more accurate than the KJV in several ways. First, knowledge of Hebrew was not very advanced when the KJV was translated. The Hebrew text they used (i.e., the Masoretic Text) was adequate, but their understanding of the Hebrew vocabulary was insufficient. Second, the Greek text underlying the New Testament of the KJV is an inferior text. The KJV translators used a Greek text known as the Textus Receptus (commonly abbreviated as TR), which came from the work of Erasmus, who compiled the first Greek text to be produced on a printing press. When Erasmus compiled this text, he used five or six very late Greek manuscripts dating from the tenth to the thirteenth centuries. These manuscripts were inferior to earlier manuscripts that were unknown to Erasmus.
After the KJV was published (in 1611), numerous earlier New Testament manuscripts were discovered--manuscripts that began to show deficiencies in the TR. Around 1630, Codex Alexandrinus (dated c. 400) was brought to England. In 1859 a German scholar named Constantin von Tischendorf discovered Codex Sinaiticus in St. Catherine's Monastery located near Mount Sinai. The manuscript, dated c. 350, is one of the two oldest vellum manuscripts of the Greek New Testament. The earliest vellum manuscript, Codex Vaticanus, had been in the Vatican's library since at least 1481, but it was not made available to scholars until the middle of the nineteenth century. Codex Vaticanus, dated slightly earlier (c. 325) than Codex Sinaiticus, is one of the most accurate manuscripts of the New Testament in existence.
At the end of the nineteenth century, two British scholars, Brooke Westcott and Fenton Hort, produced a volume titled The New Testament in the Original Greek (1881). Along with this publication, they stated their position that Codex Vaticanus and Codex Sinaiticus, along with a few other early manuscripts, represent a text that most closely replicates the original writings. In the twentieth century, many second- and third-century papyrus manuscripts of the New Testament have been discovered. Most of these papyrus manuscripts have a text that is very similar to Codex Vaticanus and to Codex Sinaiticus.
One of the most noteworthy papyrus manuscripts is P75, a copy of Luke and John dated c.175-200. It is universally recognized as a very accurate manuscript and one that bears extremely close resemblance to Codex Vaticanus. This shows that a pure line of textual transmission was preserved from the middle of the second-century to the fourth century. The papyrus P75 and several other papyrus manuscripts have helped twentieth-century scholars produce a Greek text that is even closer to the original text than that of Westcott and Hort. This most recent edition is commonly known as the Nestle-Aland Greek New Testament.
The upshot of the nineteenth-century and twentieth-century discoveries and publications of better Greek texts is that we now have a Greek text that is far more accurate than the Textus Receptus. Subsequently, English versions (like the NLT) that are based on the better Greek manuscripts are superior to the KJV, which is based on the Textus Receptus.