Well my Dixie's caught a few mice. But I don't think I'd let her tangle with a rat. They can be pretty mean - and carry diseases. JRT's are a lot bigger than even big yorkies. Dougal just hasn't got the "kill" instinct - he just barked at the mouse - but Dixie was right in there - she loved it. She also "kills" all her soft toys - by shaking them.
I don't want to start an argument or anything - but in an environment where you don't want to lay down poison I think using dogs is a pretty good way to control rodents. When I worked in a stable they had a little dog (JRT / Mini foxy) that they'd send in to the straw bedding - tho usually it was mice not rats that were the problem. Rodents can be a huge problem in stables because there's so much food around and you can't really lay down poison because the horses might eat it.
I think if I was living in the country I wouldn't be too bothered by Dixie keeping the mouse population under control (like Scotland - we don't have rabies). Loads of farms in NZ have a few cats around to help with keeping the rodents under control. In some ways it's almost more humane - a dog will kill a rodent a lot faster than most poisons (which can be really painful too
) and also faster than a cat - they tend to play with them first
.