The sharpening of tungsten tips by sputtering with neon ions shows a surprising phenomenon when the tip radius is smaller than about 20 nm: the gas ions produce a neck in the tip which becomes thinner until it breaks due to the electrostatically induced stress. This decapitation phenomenon limits the attainable tip sharpness to a radius of about 4 nm. The process can repeat in a regular, oscillating manner and is probably at the origin of the unreliability often mentioned in connection with this sharpening method. The effect can be used to determine experimentally the theoretical shear stress of refractory metals.