It is, of course, impossible to say when the Roman Empire started to fall. This is because it is hard to define what caused the fall of the Empire and when those processes started.
The Roman Empire is generally said to have fallen in 476 when a barbarian named Odoacer became the Emperor of Rome. However, many people argue that the Roman Empire continued to exist as the Byzantine Empire.
You could say that this process started in the 360s AD when the Empire was split between east and west. This led to a decline in the prestige of the city of Rome. This process was accelerated in 404 when the capital was moved to Ravenna for a while and was further accelerated when Rome was sacked in 410.
Between centuries I and IV, the boundaries of the Roman Empire had suffered only minor modifications. Constantine's decision to found a second "Rome" divides the empire into two parts (Western and Eastern). Conversion of Constantine to christianity provided on the other hand, the hegemony of the new religion. Fall of Roman empire was caused by migratory peoples .Beyond the Rhine and the Danube the germans attacked, divided into tribes and categorized as barbarians by the romans. With ever-increasing price of a military effort, the latter succeeded in general, to keep the distance. The more turbulent in the West, were franks. Others headed to the east: the visigoths to the plains of the Lower Danube, in the northern Black Sea the ostrogoths.
Around the year 370, Huns, Turkic-Mongol mixture of populations, oust the Ostrogoths and Visigoths to the west, requiring shelter Romans, setting up the right bank of the Danube. German invasion of the Huns cause increased pressure on the empire. In 406, Vandals, suebs crossed the frozen Rhine and invaded Gaul. The event is accompanied by significant turns. Franks, burgundians and alamanii captured the left bank of the Rhine. Legions of Britain were leaving over the English Channel and the island quickly and permanently was left out of the empire. Suebs and vandals who invaded Spain split the country into several kingdoms.
Visigoths rejected from the Balkans by the Eastern Roman Empire, enter into Italy and then under the roman service was launched the reconquest of Spain. Visigoths slaughtered the Vandals and the reward for their efforts was the province of Aquitaine in 417.
Subsequently, the situation stabilizes in a certain extent. Franks (in Belgium), burgundians (in the region of Lake Leman), the visigoths and suebs being installed within the empire, with the status of `federal` (endowed with a certain territory in exchange for military service) but is practically , independent. In 428, Vandals start conquering Roman province of Africa. Huns, tenuous, led by Attila, enter the Gaul in 451, but are rejected by a coalition of romans and germans.
Most people don't agree on when the Roman Empire actually fell.
Begining in the 3rd and 4th centuries A.D. the Roman Empire suffered heavy invasions from the Visigoths and other gothic tribes. In the beginning of the 300's the Roman Empire was split into three presectures. Although the empire was reunited under Diocletian and Constantine in the year 330 it never achieved the power or status it once had. The final split between the eastern and western Roman Empire solidified the fact that it would never once again be as great of a world power as it once was. This ultimately lead to the fall of the Empire.
This was just one theory. There are multiple theories as to why the Roman Empire fell.