mark,
I won't say I know exactly what's wrong with your African Grey but my Wei Wei have this same behaviour when I first have him; and yes just like you first thing that cross my mind was perhaps he have a illness.
I don't know exactly what cured him of it but I instintively approach any Pet problem the way I look at Dog Behaviour Problem; scrutinise and improve on - Environment, Diet & Emotion.
Environment - Keep his cage clean and in a well ventilated place. Give him a shower to remove the dust (African Greys produce this) say twice a week on a Sunny day. Dust are the biggest culprit to cause nasal and respiratory problems.
Diet - Ensure that the bird's food, whether seed mix, pellet or fruits and vegtables are always fresh. Include multivitamins and cod liver oil drop.
Emotion - Play and talk to the bird more often. Having the bird in the living room where your family spend most of your time is beneficial in more ways than one instead of at another part of the house.
Just sharing what works for me.
Frankie Koh