I disagree with Caregiver. I think by "breakpoint" you meant a point where the asthma improves.
Are using your rescue inhaler before you start playing?
I think you should talk with your asthma specialist. If y don't have one, get one. You should be able to play basketball safely, but a specialist will know all the ins and outs of making it work safely.
Make sure you're well hydrated before you start playing. Lungs need lots of moisture to function well.
First of all, you must recognize that chronic asthma is a life-threatening disability. By fooling around "close to the breaking point" you risk death. A tolerance may be built around the "emergency" inhaler. At some point it may not work. If there is not an EMS unit available to endotracheally intubate you you will die. In the general scheme of things, basketball is meaningless. As you grow older the asthma problem may become less severe. Only time will tell. If your "asthma does not seem to be resolved", avoid activities that bring your lungs to this point. This is nothing to "experiment with". The time to develop a spasm that complete closes the airway may be very short, and the fact it was not short in the past does not mean it will be short in the future. The best course of action is to avoid any activities that cause an asthmatic attack.