Why is diamond structure not bravais lattice? Why is diamond structure not bravais lattice? Is it because of interpretation of two f.c.c. structures?
 A: 
Is it because of interpretation of two f.c.c. structures ?

Yes, and consequently the atoms don't all have an identical environment.
A: Diamond structure is a FCC bravais lattice with two carbon atoms per site. As the OP says, it is like two FCC lattices with the second lattice shifted by (1/4, 1/4, 1/4) in reduced coordinates from the first one.
A: The definition of a Bravais lattice is "an infinite array of discrete points with an arrangement and orientation that appears exactly the same, from whichever of the points the array is viewed." [Ashcroft, Mermin, Solid State Physics, page 64].
Coming across a diamond lattice structure, the first thing we would try is "dropping" a lattice point at the location of every carbon atom.

However, if you tried to make every carbon atom a lattice point, then you break the definition of the Bravais lattice, specifically that bit about appearing exactly the same from whichever of the lattice points the array is viewed. The "view" from the carbon atom in the bottom left corner is different from the carbon atom immediately to its top right. They have different nearest neighbors.
We solve this by taking a well-worn Bravais lattice, the face-centered cubic (FCC), and saying that every lattice point there is a two-point basis. The basis is what you "glue" to a lattice point so you can distribute atoms or molecules about or on that point. Just to be clear, lattice points define the geometry of the layout. They are the locations where everything looks the same from. The basis is the thing or collection of things that repeat at every lattice point.

Now take each lattice point in the image above, and glue the following two-point basis on each lattice point. The first point of the basis is a carbon atom that lands directly on top of each lattice point, mimicking the original FCC.The second point of the two-point basis will be another carbon atom, a distance $\frac{a}{4} \hat{x} + \frac{a}{4}\hat{y} + \frac{a}{4} \hat{z}$ away from the first basis point.
We've now constructed the diamond structure as a face centered cubic with a two-point basis. You can not say the the 'diamond lattice' is a Bravais lattice, because that would break the definition at the top.
