I had the same problem.
On my set (KV-27V26) there is a circuit board that plugs directly onto the back of the tube. The connector can get worn or loose and cause the picture to fail. I fixed the problem by pulling the circuit board (be gentle when doing this), cleaning the pins that fit into the connector, and applying a small amount of silver grease. I then re-installed the circuit board.
You will have to open the case of the TV to do this; please read some of the other posts regarding safety. A TV can give you nasty electric shock even when unplugged.
You can find silver grease (aka thermal grease) at any computer store.
Arcmassif
August 2006