Can I run bloomz-560m on an RTX 3060 12 GB?

Yes

Its published weights (1.12 GB) fit in the 10.8 GB an RTX 3060 12 GB can use, with room for an unknown amount of context.

hf download bigscience/bloomz-560m

Files and context on RTX 3060 12 GB

Each file of bloomz-560m (and of GGUF versions the site has analyzed), and how much context fits next to it, with the context cache in 16-bit or 8-bit.

bloomz-560m on other hardware

Popular models that run on RTX 3060 12 GB

Assumes 90% of the card's memory is usable and at least 4K tokens of context for one conversation, with the context cache in 16-bit unless noted. Fits leaves at least 20% of that memory for context; Tight leaves less.