I would have to consult an expert on Jamestown and early English colonial efforts to know exactly what the colonists kept in their armories, but by 1604, the longbow would have been well on its way out in England. Although longbowmen could still be found in militia musters of the late 1500s, their use was increasingly derided as militarily useless and not considered much of a viable alternative to firearms past 1560 or so. While plenty of individuals likely kept bows for recreational or hunting use (traditions reinforced by law and custom over centuries don't die easily), there would be very few people capable of using the bow as effectively as soldiers had in previous eras. It would be unwise to rule out the idea that an individual colonist brought along his bow, but there would certainly not be a concern that Native Americans would copy it. There was no perceived threat from a weapon that the English themselves had almost entirely abandoned.

To reinforce what /u/MI13 said, but for the other side of the equation: the bows used along the east coast were generally superior to the bows that Europeans were bringing over with them. Early in Jamestown history, for example, there was a friendly exhibition of Powhatan and English weaponry and skill. While the English arrows stuck in their targets (wooden target shields used for practice), the Powhatan arrows went through them. After this, the English brought out their firearms, which also went through the shields. Oops. I remembered this last part incorrectly. Please see the quote below.

The Apalachee in northern Florida also had famously formidable bows. The bows were described as being nearly as tall as the archer (and the Apalachee themselves were, on the average, taller than their Spanish contemporaries). When the Spanish confiscated one of these bows and tried it for themselves, their archers couldn't even pull the string back to their faces, while the Apalachee regularly drew the string back to their ears. Two or three layers of chainmail was insufficient protection against their arrows. If the arrows didn't punch straight through the mail, they had a tendency to split and their arrowheads shatter, at which put the shrapnel made its way through the mail instead and still did considerable damage. To counter this, the Spanish started adopting the cotton armor of Mesoamerica, which was better at absorbing such attacks. These bows were accurate at 200 paces and could be fired 6-7 times while the Spanish were reloading their firearms - allowing the Apalachee to fire a volley and retreat out of range before the Spanish could respond.

Where did you see this claim? The long bow had been pretty much totally supplanted by the gun for military use by the time the English were establishing colonies.

To supplement what has been said so far, I'd like to add that if you have access to journals, I'd like to recommend reading the following article about arrow wounds caused in battle with native americans:

Bill, Joseph Howland (1862): “Notes on Arrow Wounds,”, in: American Journal of Medical Sciences, Oct. 1862, Vol. 40, Issue 88. p.366

If you don't have access, there's this article here that sums up most points:

http://allthingsliberty.com/2013/05/battle-wounds-never-pull-an-arrow-out-of-a-body/