What I personally experienced is there are some individuals who can identify specific notes down to the unit frequency (I played a 439Hz tone, the person said "Uh you're a hertz short" and I fixed the bug in my program). That level of ability is generally believed to be not learnable after the brain loses a certain amount of plasticicity.
Continuing from my own experience, people who do not have absolute pitch at that level can improve their skills in pitch detection including: identifying intervals, identifying octave, identifying common notes in an octave, and people like me who can barely tell you the interval between two notes can improve their pitch detection somewhat.
Whta is rarely or never observed it people with relative pitch gaining perfect absolute pitch after growing up, regardless of the amount of training.