This idea came to me while reading GOT. Arya's sword needle is what inspired this question. How thin/narrow did swords get throughout history and in particular the medieval period?
A sword to look at is the estoc (French) or tuck (English). This is a longsword with a blade designed for thrusting, to the point of having no sharp cutting edges, in order to defeat armor.
Sources: Alfred Hutton, The Sword and The Centuries
Egerton Castle, Schools and Masters of Fencing