It's interesting that the Dalai Lama stated "after twenty we begin to grow old" -- for the majority of us, that seems to be when life barely begins itself.
Perhaps he's being more literal and he means (roughly) peak physical maturity. Although I've read different estimates - I seem to remember people mention somewhere in the 20s for when your body has completed it's full physical growth cycle.
I'd also argue that full intellectual maturity comes well after age 20. The intensity may be higher earlier on, but maturity, as I've witnessed it, seems to peak later (assuming it does develop at all).
(Edit to add): It's been known for a while that physiological maturity of the brain occurs after age 20:
Neuroscientists now know that brain maturation continues far later into development than had been believed previously. Significant changes in brain anatomy and activity are still taking place during young adulthood, especially in prefrontal regions that are important for planning ahead, anticipating the future consequences of one’s decisions, controlling impulses, and comparing risk and reward. Indeed, some brain regions and systems do not reach full maturity until the early or mid-20s.
A recent study by Lebel and Beaulieu (see below) reinforce the above findings that the human brain doesn't stop developing at adolescence, but continues well into our 20s.
My "if it develops at all" comment has more to do with theories of cognitive development -- higher levels of functioning are fairly rare. Jean Piaget's work is recommended: