Other groups from the Middle ages who wore tall pointy hats!
In trying to discover the source of the hennin, I discovered some photos of others wearing comparable hats, which may or may not have contributed to the popularity of the fashion. One of the more interesting personages who wore a hat similar to the hennin was the Pope.
www.siue.edu/COSTUMES/PLATE15AX.HTML This drawing shows him wearing something somewhat different to the mitre we are used to seeing. I do not know if the cardinals and bishops wore the pointy hat as well. But, in some places, Jews were known to wear them.
In other medieval countries, there were groups who wore tall hats similar to the hennin. However, these groups were made up of men.
Ottoman Janissaries "wore tall headdresses, with a fold of cloth down the back. This was in memory of a revered holy man Hadji Bektash, the founder of a mystical order revered by the corps, who cut a dangling sleeve from his fur mantle and gave it to an officer, who put it on his head." Quote from the book, Everyday Life Through the Ages published by Reader's Digest.
http://www.ucalgary.ca/applied_history/tutor/islam/empires/ottoman/janissaries.html links to info about the Janissary Corps; who were Christian captives from conquered territories, educated in the Islamic faith and trained as soldiers, becoming the elite personal army of the Sultan.
The Whirling Dervishes trace their origin to the 13th century Ottoman Empire. Their costume included a tall hat like the one worn with the hennin. The modern hat is similar to a truncated hennin, but without a veil.
See www.dankphotos.com/whirling/index.shtml for photos of practicing Dervishes.