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There are a few different factors here, and this wonderful tutorial covers a good portion: Powering Portable Projects: Batteries - Tutorial Australia
Another few quick questions, does your battery pack connect them all in series? and how did you have the servo’s wired up? (As the batteries discharge they might move out of the working range of the servos depending on the type of battery)
Without any load I’ve had a battery pack like that powering 5 servo’s with little to no load on them; if you are trying to pull heavier objects itll pull more current and the voltage across the batteries will drop a bit.
The battery draining one is interesting. It will depend on the capacity of the batteries (Ah) and how much current you are pulling from them (A), say we have a battery with a capacity of 500mAh, and need 1.2 A to power some circuit.
that would be 500 /1000 (converting to Amp hours) all divided by 1.2A
= 0.4 hours
You should take a bit off for safety though. This online calculator takes care of everything as well