Asthma
The word "Asthma" is not quite the show-stopper everyone is making out here.
The outcome of an appeal will depend on the severity, frequency and nature of the "attacks" you had x years ago as well as any treatment administered.
As someone has already said the Docs of yesteryear were very happy to write Asthma, when today the same episode could be classed as a chest infection.
I had a problem with an "Asthma attack" on my medical reports aged 9 which was treated with a nebulizer (successfully). I had no further episodes or treatment. I had to locate the actual Doc from back then and ask her to change her diganosis to chest infection. This only worked as the evidence and lack of re-occurrence supported this.
Anyway, it worked and I'm now flying fast-jets.
If you want anymore info please PM me.