Yes and no. A battery's capacity isn't the only variable. You also have internal resistance. If you have a worn out to hell battery where there's dendrites on the anode and you've basically electroplated the cathode the internal resistance is going to be much higher. So if you draw the same current on a worn battery and a new battery the voltage drop is going to be way higher on that worn battery. If you're already at a low state of charge and you vdroop your way below 3.2V under the current load that battery is going to cut the current. Most people notice it as an Android phone that randomly reboots at low battery.