2 heart attacks under the age of 20 would indicate some sort of congenital abnormality or cholesterol issue - both bars to entry. Heart attacks at that age are vanishingly rare, so I suspect your mate is full of shit.
Asthma - really, this question has been answered so many times I'm not sure if I can be bothered answering it again. So I will do the search for you and cut and paste a previous answer, just to prove that it can be done.
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If you were given an inhaler at 13 (don't know what age you are now) you will probably initially fail the medical until you can prove you are not asthmatic now. This will involve using the aforementioned peak flow meter a few times a day for a month and recording the values. It sounds harsh, but it's the only way we can do it. You will need to see either your GP or practice asthma nurse to get the equipment.
This sounds similar to your situation. So - are you asthmatic now? Do you get short of breath or cough when you have gone for a run? Do you have audible wheeze on your chest? Do you get bad chest infections? Ultimately, we will require a month of peak flow readings showing you don't need medication to convince us. See your GP or asthma nurse.
Childhood asthma usually clears up by the age of 8, but sometimes drags on a bit longer.
Janner - you are right, but basic training is great for weeding out asthmatics from normal people. Conditions discovered during basic training which are obviously longstanding (like asthma) lead to immediate discharge (ie within days). In addition, the individual will have made a fraudulent application and, in my opinion, should face the appropriate consequences for that too.
I had asthma until about 16 years ago. I had my medical yesterday and the procedure for me has been for the GP from the medical centre to contact my GP about my background with the condition and to self test twice a day for 2 weeks on a peak flow meter.
I also find it odd, however as AG has noted, if you have a history of asthma and lie at this early stage to get in, I assume the level of Navy PT enforced during basic training will bring on any attacks/wheezing so you will be found out sharpish. At which point your integrity is questionable so who would trust you at that point.
I have undertaken officer selection with the army and it was flagged up then. However for that I undertook a lung function test (peak flow, extended type on a bike with a oxygen mask on and then a peak flow test again). I don't know which method is more accurate (although I can appeciate different time frames and physical requirements attached with the different services might make for different testing methods).
AG, any input (out of pure curiousity)?