Actually, most (all?) batteries have their rated capacities based on 20 hours rather than 1 hour. Therefore, a 2000mAh battery is actually rated to deliver 100mA for 20 hours, not 2A for 1 hour. If you try the latter you will probably see a significantly smaller usable capacity. Compare the graphs I posted earlier, which only go up to 1A.
(This is particularly crazy for things like car batteries, where (say) a 40Ah capacity is still based on the idea of drawing 2A for 20 hours, despite the fact that the battery is really designed to deliver hundreds of amps for a few seconds.)
However,