As far as I know, what Lars Andersen, has more in common with stage performance then with bow shooting. In order to achieve such speeds, his bow has extremely low draw weight. I can't bring any scientific proof, but I think it's biomechanically impossible even for a fit human to achieve speed, accuracy and projectile velocity with a bow to be on par with contemporary firearms.
There are additional problems with bow and arrow as compared to firearms in the modern battlefield conditions. The whole configuration itself is quite bulky - bows are pretty long (even the short ones), arrows are long too. It's much harder to shoot a bow from cover - you need to take a specific, comparatively exposed position and need space to spread your arms.
Additionally, if we are speaking about battlefield usage, although arrows may have a chance against soft armor, they pose very little threat to trauma plates.
So, the only scenario where bow and arrows can be useful is ambush (or assasination, as AlexP says), where your target isn't armored, doesn't expect an attack and you have at least one chance to loose a heavy broadhead arrow to disable him quickly. If your targed is armed or has bodyguards, the chance of follow-up shots are very low. If you have multiple opponents, you're better to have numerical advantage.