Though attacking someone in body armor with your bare hands does NOT seem like a good idea. Especially when he has a gun.
Still, the saying about bringing a knife to a gunfight isn't always true. Depends on the range. When the knife is close it can be as deadly as the gun. :-)
from all my training over the years, if someone gets within 7-10 yards of you with a knife and you don't have your weapon aimed and ready to fire when they move, you WILL get cut. If you do have it aimed and they're within 5 yards, you're still going to get cut, just as you fire...that distance can be covered within your reaction + decision time. It takes at least a quarter second to even realize they're making a move let alone decide what to do then initiate your reaction.
Still, the saying about bringing a knife to a gunfight isn't always true. Depends on the range. When the knife is close it can be as deadly as the gun. :-)