There is little energy left in a 3 V lithium battery below 2.7 V, so it was somewhat satisfying when I found that the battery from one of our remotes was at about 1.5 V just after it had stopped working. That remote had been quite good at scraping the energy barrel.
Devices differ enormously in their ability to work from batteries that aren't new. Some devices will be spectacularly good, like that remote, and other will stop with a battery that is still more than 3 V. Little LED lamps running on CR2032s or similar are often very poor, in that the light output starts dropping immediately, and while they will run for ages from a battery at 2.8 V, there is next to no light output with the low voltage.