***** NOT A TEACHER *****
Hello, O.P.:
They say that in English (and in other languages?) context is king.
Therefore, perhaps both sentences could make perfect sense.
Teacher: Last week, I told you students to wear a T-shirt to class today. I notice that everyone is wearing theirs -- except James. What's the problem, Jimmy?
James: I'm sorry, Mr. Smith. But I have been so busy that I haven't had time to go to the store. I promise you: I am going to buy my T-shirt tomorrow.
*****
Teacher: I told everyone to wear a T-shirt today. Why aren't you wearing one, Jim?
James: Believe me, Mr. Smith. I bought my T-shirt last week. I had planned to wear it today, but the dog chewed it up, so I have to buy another one today.
James
NOTES:
1. In 2014, most people no longer say "Everyone is wearing his." So we have to say "Everyone is wearing his or hers" or "Everyone is wearing theirs."
2. Probably in real life, it would sound more natural if James in the first dialogue said, "I am going to buy it / mine tomorrow."
3. In the second dialogue, probably "I bought it / mine last week" would sound more natural. In other words, the teacher has already used the word "T-shirt," so there is no need to repeat it in ordinary conversation.
4. James can say "the dog" because he is thinking of a particular dog -- his dog.