It depends. In many cases involving experience-, item-, or mood-based evolution, no, age and evolution are separate concepts. You can have a newly hatched pokémon that quickly evolves due to the care it receives, or you can have a very old pokémon who’s stayed in their base form all their lives. Evolution is merely a response to outside stimulus in most cases, whether that stimulus is an increase in battle experience or exposure to the right climate or so forth.
However, age can be linked to evolution. In so-called “trade-based” evolutions such as alakazam, gengar, and so forth, the ability to evolve is unlocked over time. That is to say, it’s not just a matter of experience. It’s a matter of so much experience that it’s impossible to reach that point until an individual is advanced in age. (Alternatively, exposure to the trainer editing process of a trade machine can also unlock these pokémon’s abilities to evolve, but the point is, under natural, normal circumstances, a kadabra may be highly advanced in age before it’s ready to evolve.)
The Pokédex According to Bill 