Do a search on this forum on asthma - you might not be too happy with the result, as it seems that the RAF have been fairly categoric in the past. The RN may have been easier to get in with a history of childhood asthma. The general opinion seems to be that you go first to a private specialist who reviews your notes/history and says either that you were mis-diagnosed in the first place (possible, in fact) or that the bouts you had were so long ago and so few as to be insignificant. You then present this opinion with your application.
I'd expect that a four year period with no symptoms would not be enough. That would imply that a 20 y/o would have had his last attack at 15 ish. More like ten to twelve. Asthma did become "trendy" amongst doctors about 25-30 years ago - possibly to cover their backsides, and there is a suggestion that modern pollutants can cause asthma or asthma-like symptoms in children, but that they improve/recover in their teens.