I was actually thinking about this yesterday when the first draft of a graph I was making looked nifty, so I removed the numbers and saved a clean copy[1]. There are plenty of issues data-wise: the colors repeat, lines appear and disappear for no reason, without numbers of labels you can't draw much of anything from it, but I found it to be inherently pretty despite the failure to communicate much of anything effectively. I had to tweak it quite a bit[2] to get across the information I wanted, and it definitely loses most of the visual appeal (at least to me, I also accidentally have 2015 in the legend on that revision despite not having 2015 data on the graph).
That's not to say you can't mingle art and information, I just think it's much more difficult than producing either on its own.