This was the first book in a series but I only read the one book.
At the start of the story the main character is living in a village or small town. He travels to a city for training and gets tested (he may have been tested at the village, I can't remember exactly) and is discovered to be a mage. I think most mages at the school are from wealthy families.
There are mages and archmages, I think generally mages are weaker and when mages run out of magic completely they lose their magic permanently. Archmages can run out of magic without losing their magic permanently. No archmages have been born for a long time but mages can form a link with another mage and become an archmage.
Near the end of the book the main character attends a party with a friend from a wealthy family and cools the room down to make people more comfortable, then he gets attacked and he runs out of magic since it is all used to heal himself keeping him alive. This happens at the end of the book and I think the implication is that he is probably an arch mage but this wasn't explicit.
I think I read the book about 4 years ago, I think it was a fairly modern book but I wouldn't normally pay much attention to the publication date.