There wasn't a single point. The empire lost control over it's territories gradually, losing power and relevance step by step. With one big resurgence along the way.
- 370s large groups of Goths fleeing from the Huns enter the empire. Unlike other barbarians they never fully recognize the authority of the Emperor, and continue living semi-independently in various areas of the empire.
- 395 the split into Easter and Western empire unintentionally becomes permanent. There was never any intention to fully divide it, but the position of Emperor became inherited and Theodosius had two sons (both children at the time)
- Because Emperors were children they become figureheads while the actual power is held by other people (advisers, regents, military commanders). It continues once emperors grow up. In the East this works out, but in the West it results in the collapse of the central authority
- During early 400s the state apparatus in Britain and northern Gaul becomes especially weak and various groups start forming their own "states" still owing fealty to the Empire (and even fighting for the Empire most of the time). Groups of barbarians even reach Africa and establish themselves there.
- 476 - the traditional date. The last Western Emperor is deposed and nobody put in his place, because the position was irrelevant by the time. About this time other semi-independent groups just continue acting on their own
- 490s - the Eastern empire continues on, and they want the West back, so they authorize a group of Ostrogoths led by Theodoric to take over Italy. The Ostrogoths establish their own kingdom, still nominally claiming to be the Western Empire. In pratice fully independent.
- 530s-540s - Justinian decides to finally reclaim the entire Western Empire. This if fairly successful - Italy, Africa and parts of Iberia are reclaimed. However it coincided with the first know instance of bubonic plague which throws everything into chaos and prevents the full reconquest from taking place.
Personally I'd say for most inhabitants of the Western empire, 420s were the turning point. Their immediate rulers became non-Roman, even if in theory they were still under Roman command.
Edit: this is all about the Western empire. East is a longer story and will continue until 600s, 1204 or 1453. Which is a whole another debate.