I saw no measurable difference when I left Twitter in the traffic to my site and neither has NPR. I think that Twitter, and possibly all social media, has got an unearned inflated importance in how much benefit it actually brings in terms of traffic to a site.
My top referrers over the last 30 days are:
If I look at January of this year when I was still on Twitter and sharing my content to the service…almost nothing changes. Twitter makes it into the top 20 but it only sent my 17 visitors compared to the 200 both Eleanor and Elizabeth sent me in the same time period.