It is unlikely that the player will run short, unless they need to save very frequently for some reason. However, there is another use for Rings, which would run out the supply much faster: they can also be used as a means to Buy Advice.
Be advised that Advice usually refers to the current mission, and not parts of it, so if there are multiple tasks that require completion, and one of them is completed, the game may well consider that the bulk of the mission is complete, and default to saying something equivalent to, "the player does not need advice at this time and should not ask".
A mysterious old man assists the Nomad Soul by offering use of Rings of the Art. The player is not told until the end whether this is an unknown person, or Matanboukous, or perhaps his apprentice, Assam.