Hi Jean (and Bird, if you're still milling around),
That's what was happening to my guy Jake - as you can see in this thread about it: http://www.pets.ca/forum/showthread.php?t=46076
Turns out he was suffering from something simliar to an URI - it looks like he had an outbreak of Feline Herpes, which he caught off of our kitten, Alley. According to some research I've done, about 90% of cats have it, but you usually don't see any signs unless they have experienced stress, change in environment, etc. The coughing is due to congestion - you might notice your cat swallowing a lot more too - trying to swallow down the phlegm.
I would watch your kitty for signs of not eating as well - when their noses get stuffed up they don't want to eat as much.
You can pretty much just treat the symptoms, and everything should wrap up eventually. Jake just wrapped up a bout of Clavamox, he's on some diuretic pills to clear up some fluid in his lungs and has some L-Lysine in his food, and he's almost completely better.
I would ask for a check for this. If not, you might need x-rays to see if your cat has something more serious, like asthma or bronchitis. How old is your kitty? Asthma usually appears in younger cats, but bronchitis can happen any time.
Hope that helps!