Chronic inflammatory demyelinating polyneuropathy is disease believed to be due to immune cells, cells which normally protect the body, but are now attacking the nerves in the body. As a result, the affected nerves fail to respond, or respond only weakly, to stimuli causing numbing, tingling, pain, and progressive muscle weakness.The likelihood of progression of the disease is high. This study is designed to examine whether treating patients with high dose cyclophosphamide (a drug which reduces the function of the immune system) and ATG (a protein that kills the immune cells that are thought to be causing disease), followed by return of the previously collected blood stem cells will stop the progression of CIDP. Stem cells are undeveloped cells that have the capacity to grow into mature blood cells, which normally circulate in the blood stream. The purpose of the high dose cyclophosphamide and ATG is to destroy the cells in the immune system. The purpose of the stem cell infusion is to evaluate whether this treatment will produce a normal immune system that will no longer attack the body.