There has been a lot of guidance on this matter, but the bottom line is if someone knowingly gives the wrong information on joining it is classed as a false attestation. Now I have not heard about someone not declaring health problems, but I have heard of personnel who have failed to declare criminal convictions and they have been discharged.
As for the Asthma, if it develops once you are in, and is only mild you get a Med Cat of G2. G2 is a medical note and does not affect your career at all. However, if it is anything other than mild you can get a Med Cat of G3 and this could lead to discharge for an airman or no promotion for an officer.