When I ride very hard (all out TT type effort) I do get a hacking cough going afterward, especially in the early season, before my lungs get used to it. It seems to go away or at very least lessen after a while. It's pretty normal to hear a lot of folks coughing after a race. I don't really think it has to do with fitness - even the pros get it - Phil called it Prologue Cough in the Paris Nice telecast.
Of course if you already have asthma I'm sure it needs watching.