Typically, just when their bodies start to break down. Cataracts. Arthritis. Failing organs. Weight loss. Slowing down.
My mitred conure is 19 years old and he is my oldest *confirmed* age bird, but I may have some others older than him. I just don't know. Conures have an estimated age of 30 years, and I know of one estimated to be 40 years old and a quaker that may have died when he was 50.
In fact, a member here has a cherry headed conure that is 31 years old!
Charlie, my mitred, isn't showing his age... but I do have a bourke parakeet that's 16-18 years old (expected lifespan is 11-15 years, heard of one that died at 21[?] years of age and one still living at ~24 years old!) and she's definitely showing her age! She's having more health problems than she did when she was younger, she's not as active (no longer runs around on the cage bottom), has lost weight and she has cancer on her right foot. I think she might be developing cataracts but I'm not entirely sure yet. (they just appear lighter than they used to be)
I used to own a cherry head, he was estimated to be about 17-19 years old when he passed away, but I would not be surprised to find out if he was in his twenties or older. He couldn't fly, he did develop cataracts, had a less than healthy kidney (I suspect kidney failure), had trouble gripping with his feet, his legs were actually smooth (rather than almost scaly like).... although for all of his problems, I don't know how much I can attribute to age and how much due to a ferret attack years before I got him. Cataracts were definitely due to age, but I don't know about the rest.
I also have a cockatiel that is over 15 years old (I only know his history starting in 1998, and he was an adult then! Not a clue how old!), and tiels have an estimate age of 20-25 years, but have the potential to reach 35/36 years old. So far, he's not showing any signs of aging! He is thin, but he's always been thin!
It really depends on the birds themselves. Some may reach their expected lifespan and maybe even surpass it without showing any signs of aging! Indian ringnecks for example have an expected lifespan of about 30 years old, and I heard of a hen still breeding at 31 years old!