In that case, what if somebody grabs that person with a thick, dry quilt or blanket?It depends upon the voltage.
Cloth has air gaps through which an arc can form.
The breakdown length for an arc on a flat surface is about 3kV/mm thus, for example, 1700Vac with a 2400V peak voltage will jump about 0.8mm.
If you can grasp loose dry clothing where it's not touching skin, then you likely would be safe.