Training uses gradient descent, so you want to have good precision during that process. But once you have the overall structure of the network, https://arxiv.org/abs/2210.17323 (GPTQ) showed that you can cut down the precision quite a bit without losing a lot of accuracy. It seems you can cut down further for larger models. For the 13B Llama-based ones, going below 5 bits per parameter is noticeably worse, but for 30B models you can do 4 bits.

The same group did another paper https://arxiv.org/abs/2301.00774 which shows that in addition to reducing the precision of each parameter, you can also prune out a bunch of parameters entirely. It's harder to apply this optimization because models are usually loaded into RAM densely, but I hope someone figures out how to do it for popular models.