Some do, some don't
It all depends on where they base their morality. If their morality is based on the needs of some religious institution, i.e. RCC or JWs, then they don't often have all that high a morality. If they are based directly of Jesus' teachings, or I might even add Buddha's from what I little I know of his teachings, then their morality will usually be higher. One poster already noted the high morality of an isolated Russian Christian group who only had a few examples of Jesus' sayings to go on for determining their morality.
I've met non-religious folks with a much nobler sense of morality than most religious. But when I enquirer as to where that sense comes from I usually find that it goes back some sort of religious basis formed early on in life or in their parents life and passed on to them. Others noted how religiously based morals made some associates life better and simply emulated what they saw worked.
Forscher