the cart cannot run dry. there is extra in it to prevent exactly that. The printer declares a cart full empty even though it actually has about 10mL remaining and the motivated student could harvest it.
Actually, when the printer malfunctions, the cart can go dry.
The printer has no way of measuring how much ink is in the cartridge. It knows how much ink the cart starts with, and tracks how much ink it has tried to print. It declares a cart empty when it has requested the rated 80ml.
When the black valve malfunctions, black ink leaks out without the printer knowing. The cart can be physically empty, while the printer still thinks that it is mostly full.
This is a common failure mode of the 3880. The symptom is no black. To verify, weigh the black cart. If it really is empty, you have a stuck valve.
You can try to get the valve unstuck by switching to the other black and back. If you are lucky, this (and a new black cart) will get you back up and running until it starts leaking black again. If that doesn't work, buy a new printer.
The other failure that I frequently see with the 3880 is the failure of the networking port. While it doesn't make economic sense to fix the port, you can use and external print server as a work-around.