Well - I'm at a total loss now.
I spoke to Sharp and they told me that I am connecting it the wrong way. I'm using a VGA to DVI adapter on each end of a VGA cable. They told me that I couldn't be as VGA is totally different from DVI. So I explained that the DVI connector on the TV supported DVI-D and DVI-A as it had PC Digital and PC anologue settings. The DVI-A is just a pin mapping of a VGA signal.
It this point he told me that DVI-A doesn't exist

and that I should be using either DVI-I cables or DVI-D cables. He then didn't understand that a DVI to VGA converter only maps pins and he thought that it actually converted the signal

Apparently, he has never heard of any one using a VGA cable and adpaters to run their TV.
So, the only solution he gave was to buy a DVI-D or DVI-I cable and to try that one out.
So am I going mad or what? I'm sure many other people run their VGA cards via VGA cables and adapters.
Failing the DVI route, he told me that it would cost me £350 to return the TV for testing. I have lost all confidence in Sharp being able to identify a problem with my TV given that they don't even know that VGA to DVI-A adapters are all ratified in the DVI standard.
Anyone - any more advice on things to check before I potentially waste another £350
Edit: - perhaps I am going mad and its my graphics card or connectors thats the problem. But being able to run another monitor via the same cable and converters suggest that its not.