Yes it can, yes it does and yes it has. The point is that if the school is your sponsor, they, in effect, say to the government and immigration that they will be responsible for you in the event that you get into some kind of trouble such as not having the funds for paying your rent.
If you leave them before your visa expires because you are unhappy or something, they are well within their rights to inform the government that you have quit and they are relinquishing all responsibility for you as you are no longer working for them. The government does NOT like foreigners being in Japan without a sponsor and they may come looking for you and deport you.
This is a worst-case scenario, but it could, and has happened.