When our veins get cut down,the following things happens:
a) The arterial blood(blood that flows through veins) leaks out which leads to decrease in blood pressure.
b) The decrease in blood pressure also reduce the flow of blood.
c) Since arterial blood transport oxygen,minerals,nutrients etc to different parts of the body,leaking of arterial blood stops the transport of these materials.
So, it is fatal when our veins get cut. 
Veins carry deoxygenated blood from different parts of the body to the heart.if the veins are cut without any external bleeding then the deoxygenated blood clots internally.preventing the flow of blood to heart eventually leading tom death .if it bleeds externally then due to blood loss from the body the person will die