Remember - safety is the first thing.
We have a guy who plays asthma.
Since he can't put as much air in the horn safely, his range is limited (don't know how extreme his case is compared to yours) but he's still loud as crap. One of our best 2nd trumpets.
Probably, depends on how severe your case is, my cousin just aged out of the Cavaliers, yes the Drum Core, and he had asthma, and my friend Kyle also has asthma, and he said that playing the trumpet actually made his asthma better. But remember everyone is different.
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