Miss Nancy - My guess here is that it's not the Bounces that's too low, but the Samples. With no directional lights, the final random sample has to be very high to get an accurate, stable result. As the evaluation point gets farther from the area light source, or more bounces are needed to hit that source, the fraction of the sampling hemisphere the area light takes up - its solid angle - drops and most of the random samples will miss it. That's my story and I'm sticking to it until somebody tells me the right answer.

Whichway