I personally think Stefan is on to something here. If the problem seems less when using alkaline batteries it does sound like a voltage problem and nothing to do with the capacity (mah) of the battery.
If the DF takes 6 AA batteries that means a max total of 9 volts (1.5 volts each) when using alkaline batteries as apposed to 7.2 volts (1.2 volts each) for rechargable batteries. If the DF needs say 7 volts to work, it doesn't take much of voltage drop (depleting batteries) with the rechargables to freeze the camera up.
So, to Stefen's point, if your using a cheap charger to charge you batteries that only sees the batteries as one block it may not be charging them efficiently and charging one more than the other. That results in each battery holding different charge states and with each additional recharge the difference can get greater. As the batteries are used in the camera, one will deplete faster and the voltage will drop sooner.
For all my rechargables I use one of those fancy computer controlled chargers that individually monitor each batters and also allows discharging and refreshing cycles. It would be interesting to know what chargers people are using to rechrge their AA camera batteries as the normal fast chargers do not charge all the batteries equally and are known to actually damage batteries as they make them so hot when charging.
