As a halfway decent archer and a halfway decent SCA swordsman, I can tell you my thoughts: it isn't really even close. Shooting archery takes some time and patience. The default D&D bows aren't the 90-100 pound pull, English longbow things- rather, they are made for the average 10-11 strength individual- who could not pull such a beast as the Yeomen used. (Yeoman might have been 11s or so in strength overall, but the forearms, etc., were at least 18s, I'd think)
But give me a typical adult male (not to be sexist- just an example) and a 50 pound pull bow and 20 days of training every day (with a few off for muscle recovery) and he can shoot somewhat and would be somewhat dangerous to his opponent. Heck, give me an average 45 year old woman (was my mom one time, actually) with a 35 pound pull bow and 20 days of training every day (with a few off for muscle recovery) and she could plunk it. Now could she shoot into two combatants fighting toe-to-toe and hit the enemy one without hitting her own some of the time? No, but that's a feat, isn't it? And yep, I'm talking about bare bows here, none of these modern pulleys and aiming devices.
In my experience, give me a fairly athletically inclined male with 20 days of training sword and shield (with a few days off for muscle and bruise recovery) and you know what you got? Nothing much. These folks are referred to (in my area) as speed bumps. Any fighter with several years of training can kill at will and the newbie will not be able to touch the vet who is really trying.
Gotta remember, sword fighting includes defense, shooting an arrow basically does not.
In reponse to the superb archer wondering how he would do on the battlefield- I bet you would kick tail. Sure, you'd have to learn to watch the whole battlefield (instead of focusing on one target to the exclusion of everything else) and move, move, move, but that shouldn't take too long. Just leave the front row of the enemy alone if it is engaged with your front row. Shoot the enemy archers. Shoot the enemy reinforcements. Above all, shoot those dang nam spearmen and polearmsmen fighting (in relative safety) from the second row.
As to SCA fighters on a real battlefield- yeah, pretty close, but SCA doesn't allow swinging at the lower legs (knees cannot be sufficiently armored to allow it to be done safely in the SCA's full force mass combats). As archaelogical finds have suggested lower legs very much were a viable target, the SCA falls short here. Of course, folks have developed defensive styles that do not take protecting the shins into account. So, pretty close, close as we can, but not quite.
Yet another way of looking at it: During the Hundred Years War, the English peasantry were required by law to practice with the longbow. Everyone shot. The English force in France had _thousands_ of longbowmen. That just doesn't sound like an exotic weapon to me.