Why doesn't life produce diamond, graphene, nanotubes, etc? Could it?
Biology does not have too many uses for super hard but brittle materials like a diamond. The hardest biological materials are also very tough as well since they need to form complex shapes for maximum utility. Limpet teeth are the hardest biological material known but are also incredibly strong, stronger than steel, not super brittle like diamond1.

As for graphene, the big problem there is reactivity; graphene is a great substance except it is pure carbon and has a huge surface area. If you put it in a solution full of carbon reactive chemicals, it starts doing weird things, and it has particularly weird interactions with cell membranes. Structurally, it is far more trouble than it is worth in something made of cells2.

You also have to remember that, in evolution, it is generally about "good enough", not perfection. Making a pure carbon structure would be an incredibly costly process in a biological (and thus aqueous) environment. 

References: 


Extreme strength observed in limpet teeth
Exploring the Interface of Graphene and Biology