Not in the way you're thinking. This is at the time when the King had absolute or near absolute authority. King John was the ruling class. While it was the English nobility that opposed the king, they did so (1) at great risk to their own safety, and (2) as the only ones in a position to do so. Certainly no serf could have accomplished this -- they would have been killed.
Second sentence of the (excellent) Wikipedia article about it:
> First drafted by the Archbishop of Canterbury to make peace between the unpopular King and a group of rebel barons, it promised the protection of church rights, protection for the barons from illegal imprisonment, access to swift justice, and limitations on feudal payments to the Crown, to be implemented through a council of 25 barons.