No, it isn't. The GPU only supports the concept of a single index buffer being bound at a given time; graphics APIs expose this fact by only providing scalar methods for setting index buffers (that is, they only let you set one at a time).
In general this isn't a concern, although it seems wasteful at first glance. Most complex 3D models wouldn't be able to share a significant portion of their non-position vertex attributes anyhow -- things like cubes are a pathological case.
It also shows up when you're building grid planes for things like tiled backgrounds in 2D games, but in such games the footprint of your geometry overall is generally much smaller so the minor waste that occurs in the data duplication isn't usually worth noting. If it ever does become problematic there are higher-level approaches to reducing the geometry set that can be applied (such as rendering fewer tiles).
The architecture of GPUs has advanced to a point where theoretically this wouldn't be a terribly difficult feature to support in some new hardware, but the sheer lack of demand for it from any major players means the cost/benefit ratio isn't in anybody's favor.