I have noticed in Europe it is very common to run a few squares of tile behind a pedestal sink as a backsplash. Much less common in the US. Instead we'd tile the entire wall, hell, we'd tile the ceiling, and anything else we could think to tile. I presume most comments on here will say it looks like a mistake. I say it is total preference and if you like the look and it solves the problem, that is what matters. Find a tile you like and run a few squares.

I've noticed the same thing in Europe as Kendrah has - it seems to be pretty common to run a few tiles behind a pedestal sink as a backsplash. It is hard to find examples here in U.S.

I think that if the tile is the same color as your bead board - or matched the floor/shower tile like in the above photo (or blended with the color of the wall paint), it would look fine.

Picture is worth a thousand word as they say. Context is key.