First off, do all the tutorials accessed under the "Help" menu. If that doesn't clear things up read the help and look for info on bone layers, specifically Flexible versus Region "binding". Also look for anything regarding Bone Offset. This will be what you need to know to fix your problem most likely.
In a nutshell, the default binding of a bone layer is "Flexible". Every bone will influence the entire vector layer a little bit. Region Binding, points are influenced by the closest bone with the highest strength.
Your issue is due to the fact that the arm bone is close to the chest and even if the chest has a bone the two are fighting over the points that are close to them. You probably need to use bone offset AND region binding. I'm too sleepy right now to rewrite the help files in this post.
The info is in there trust me.
-vern
