The 15th century tournament sword described by King Rene had a blunt point and edge. There is a tendency to assume that this was equally true for earlier tournaments.
However, while the late 13th century English Statute of Arms prohibits the use of pointed swords and knives in tournaments, it says nothing about the edge: the permitted weapon is an espee large pur turneer.
In the middle of the 14th century, Geoffroi de Charny wrote series of questions about possible scenarios in jousts, tournaments and war. In his ninth hypothetical question about tournaments, he imagines a scenario in which a knight is pulled to the ground and his opponents cut the girths and breastplate of his saddle. If the only allowed weapons are tourney swords, what did they use to make the cuts? Does that mean that the edges of tourney swords could still be sharp when he was writing?