Exactly what you said. @emsr2d2
So "this is your playing/studying age." is unusual? I should say "you are too old to play." but it doesn't mean what I want to say exactly. @Tdol
I think we should learn how to think the way English speakers think, I can figure out why those words are not understandable to native speakers. The same wrong way of thinking can be found in this phrase 'four-age baby' (the correct one is 'four-year-old baby') or 'She has 18 ages' (the correct one is 'She is 18 years old')... Applying in your situation, I think 'age can't play', 'age can't study', or simply 'age' can't be used that way. I wonder if 'playing time' can be used? I think we should say as Tdol suggested, 'you are too old to play', 'you should play at your age', 'you should learn at your age', ... I think we can always find a way to express what we want (in English way).
That's only my share on understanding why those words are strange and meaningless. Refer it at your own risk.
Thanks!