Like Magua said, the "incredible power" of the English archers has become kinda legendary, but was in fact partly exaggerated. Facts show that the French military defeats were mainly caused by the mediocrity (or worse) of most French commanders of the period.
As Magua said, many nobles considered they could not be defeated by a bunch of peasants weilding weapons dedicated to hunting (namely: bows), and therefore totally underestimated the qualities of longbows. But there was more than mere arrogancy.
You have to realise that a knight had to pay for his horse, his weapons, his armor, etc. And that wasn't cheap. Many knights fighting in battles were not the elder son of their family, and therefore didn't own any land to pay for these war expenses. Their main income was ransoms. That explains why French knights were so eager to run straight to the melee: they wanted to capture an enemy noble and ransom him, because that was basically the only way they could earn their living.
There were also many issues with commanders not being obeyed during battles, because some of the nobles they were commanding were of higher nobility, and refused to be issued orders.
I am not saying the longbow wasn't a very good weapon, but the bad French strategy (or the absence of strategy) in front of those longbows was the main reason of the French defeat. There are actually records of battles won by France, although they were facing an army of longbowmen: French simply used a good strategy during these battles.