Maybe not half full, but yes. The generator fuel pickup is deliberately placed a bit above the bottom of the tank so it won't run the tank dry and leave you stranded.
Another common problem is the rubber fuel line between the top of the tank and the chassis, or between the chassis and the generator, can develop cracks and let air into the line. Unlike the main engine which has a fuel pump in the tank, the generator's fuel pump has to suck the fuel out the top of the tank like drinking through a straw. This is so a leak will stop flowing by itself instead of emptying the tank onto the ground.