You have another stamp in your passport, that shows you when your current stay expires. It looks like in the picture. Until that date you either have to leave the country or apply for an extension.
When you cross the border they will stamp you in for 90 days. After that you will have to get extensions of stay at your immigration. You will need the paperwork from your school.
I'm at a Thai school in Bangkok and got my visa in January. When I crossed the border I got a 90 days stamp. Before that one expired I took the paperwork from school to immigration and got an extension of stay by 60 days for 1900 Baht. Before my current stamp expires I will again have to go to immigration to apply for an extension of stay.
I arrived at the consulate at 9am and got number 472. I booked my return flight in advance at I think 9pm so even if I was the last one collecting my passport, I could have made it.