Your question doesn't say if this is in an industrial setting or more general humanoid robot.
In an industrial setting, I think you can get by with as few degrees of freedom as possible. For the most part, the environment, task, and end-effector are all custom designed and singular purpose. So if you can get by with only 3 or 4 DoF then go for it. Extra DoF don't help at all and only increase manipulator cost, weight, and moving parts to fix.
However things get more interesting with more general purpose arms. When I was designing a new humanoid robot I did a study on the best arrangement of joints. It became clear that the usable workspace of the arm was not very large. More joints significantly increases the arm's workspace. This restriction was severe enough that despite having a 7 DoF arm and a holonomic mobile base, we still added a DoF to the torso of the robot.
Also note that just because an arm can reach a point in space, doesn't mean that it will be very usable there. @leftaroundabout brought up a good point about gimbal lock, and @kucar had good points about joint limits and avoiding obstacles in the workspace. But there is also manipulability. For a given end-effector pose, depending on the arm configuration, the arm may have vastly different speed and strength capabilities. More DoF give you more flexibility in choosing the best arm configuration for the job.