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Yes, but it is not a guarantee, nor is it a simple process...but it is possible.
The first step is to visit a recruiter with your DD-214 and medical records. The recruiter can process an enlistment package for you requesting a waiver. To do so, the medical condition has to be one that (i) will not interfere with you duties, (ii) was misdiagnosed, (iii) or that was properly diagnosed at the time of discharge, but that has since subsided.
What would significantly increase your chances of a recruiter being successful in pushing through your waiver would be to get a letter from a civilian specialist that states that you do not have exercise induced asthma. That will give the recruiter something more than your word to get permission up the chain to send you for testing for the condition.
If no proof of the condition still exists, than a waiver should be granted. If, however, the testing shows you have the condition, the waiver will not be granted.
Thank you for your questions, and good luck.
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