We have rescued various birds from proud cats who were not hungry, but just showing off their great hunting prowess.....but the bird that stayed was a baby House Finch my son found when he was in 5th grade..tiny little thing--just a couple feathers.
We sighed and fed it bits of cat food and bugs...an LO! it grow! and thrive! Them nasty little finches are hardy things! It grew right up to become a female finch, and laid eggs and learned to fly around the room and help Rita type at the computer.
We swithced her from cat food to bird food at the right time, and she would eat seeds from your hand and 'try' to eat the letters from a page of typing...she loved to perch on the rims of spectacles and pluck eyebrows, too!
She had a temper, as all finches do, and bitterly defended 'her' cage against such intrusions as 'changing the water' and 'cleaning the cage'.
Obnoxious little thing lived about 5 years in captivity --making messes and endearing herself to all (except the cat, who tried a couple of times to do a "Sylvester Puddy Cat " on the cage with semi-disasterous results...then gave up.