Bob, who appeared in all of the series (albeit very briefly in series 3), is the aging host of daytime television show Country Matters, which he has presented for several years. He has an extremely bad cough - which is not only awkward, but irritating. Conveniently, when rehearsing or off the air, Bob's cough is fine.
During the second series, the cough annoyed the producers so much that Bob was given a warning to get over the cough or the show would be cancelled. However this never stopped him (despite wasting all of his cough medicine by coughing it all over his dressing room. The medicine he did manage to take, though it did clear up his cough, made him constantly fart as a side-effect) and his career grew to presenting Badger Watch where he scared badgers away with his coughing.
In the 1996 Fast Show Christmas Special, Bob is advertising a CD of folk music featuring himself and friends who have appeared on his show previously, namely "Hiccuping" Murtagh Blethyn (Simon Day), 'Sneezing' Clive Tucker (Mark Williams) and the most prominent of all, Jed Thomas (Paul Whitehouse), who can't stop saying "ARSE!" between words, presumably from Tourette's Syndrome. Simon Day also plays a character called Harold Larvie who can't stop intermittingly shouting "ARF! ARF! ARF! COCKSUCKER!", again presumably because of Tourette's. Harold and Jed appear in The Last Fast Show Ever in a sketch where some of Bob's friends (including a sneezing man who is not Clive Tucker, and a woman who constantly belches and farts) show up to surprise him as he is filming the final episode of Country Matters, which has been cancelled after twenty years on the air.
The two characters who most regularly appear with Bob Fleming, Clive Tucker and Jed Thomas, appear to swap names after the first series. Even more confusingly, on the Fast Show Live video of 1998, the two characters are introduced as "Clive Thomas and Jed Tucker".
The real-life inspiration for Bob Fleming was Jack Hargreaves.
The character name is itself a pun - Bob Phlegming.