This prophecy was certainly fulfilled in its entirety in the early church, especially in the lives of the apostles. The big question is whether "they" refers to just the apostles or if it refers to all believers in all times.
Still, even when Jesus walked the earth, He did not heal every sick person He ever saw. So, it is incorrect to assume that every person who ever got sick would be healed by the closest Christian. There certainly have continued to be times throughout history and continuing to this day when God has miraculously healed people.
Just two nights ago, I watched a DVD entitled "More than Dreams", in which five people tell their accounts of Jesus appearing to them in a dream and bidding them to follow Him, leaving their previous religion. One of these individuals from Nigeria reveals that His father was so enraged by his new Christian faith that he brought him poison to drink. The son does so, but is unaffected by it.
So, the prophecy was fulfilled completely in the apostles, but God still does such things even today--not in every case, to be sure, but definitely in some cases.