Can thoracic outlet syndrome kill you by damaging important nerves and arteries?

Probably not: Although it can cause significant disability (extremity numbness, weakness, etc.) if not attended to. Physical therapy helps in many people and sometimes removing the first rib and removing scar tissue can significantly improve more severely affected patients.

Not definitely: Thoracic outlet syndrome can have a profound affect on quality of life, by impinging on either the artery, vein or nerve near the first rib. Removal of the first rib, either via the arm pit (axilla) or neck can alleviate those symptoms.

Usually not: But, if you are compressing a vein or artery and there is a resultant embolism to brain circulation, significant risk of stroke and potential brain hemorrhage changes the situation and this could be life threatening.