Not for any definition of curvature that springs to mind. There may well be definitions for which that's true (let me know, I'm curious!), but they're not what I meant. As I said, I was considering covering a cone with physical tiles. My point was that the same tile cannot be placed at different heights along the cone, because a slice along the tile will have to conform to ever larger circles.
The Gaussian curvature is the same. That is the curvature that can be measured "from within" the surface.
What you wrote is of course right as well
Ah hah, neat!
How do you define the curvature for a cone when it has a singularity at the apex? From the definion that I know (angle defect), the tip of the cone will have curvature