Wrong. So very very wrong.
The Roman Empire was destroyed by multiple groups of barbarian tribes, which can be split into three basic categories:
1. The border Foederati, which were (((allies))) of Rome and basically served as buffer states within the empire. The most notable of these were the Franks, who later conquered all of Gaul to form a chiefdom that would eventually become modern France. The Franks were Germanic, but on the "wrong" side of the Rhine; so were forced to fight other Germanic tribes that wanted to raid the lands inside the Roman Empire; until the Franks realized the Roman Army was in full retreat from the British Isles and Gaul, so decided to conquer Gaul for themselves.
2. The migration barbarian tribes from what is today Ukraine. This included the Visigoths and Ostrogoths; who would conquer Spain and Italy, respectively. Both of these tribes were forced to leave their lands by the Huns, fought as mercenaries/foederati for a bit, then decided to loot and pillage their way into the Roman Empire's best provinces.
3. The Huns. Crazy nomadic nomads from central Asia. This group does not need to be explained.
So in short, Rome was not conquered by peasants.