Vertices should only be attached to one joint. So work your way backwards. So when doing an arm, do the wrist joint first, and attach the hand vertices to it (assuming you dont have finger joints). Then select the elbow and attach all the lower arm vertices to the elbow joint, but not the ones you just attached to the wrist joint. etc etc.
During an animation, by rotating the elbow joint, you are moving the wrist joint, which in turn moves the wrist joints vertices, so that's why you only attach vertices to one joint.
Main technique - work backwards.
Hope that helps.