I do notice buffering issues much more frequently with Tunein than with typical AV channels, even though the latter must obviously impose a far greater load on my equipment and internet connection. Moreover, I frequently experience the same problems when using the Tunein app on my Android device, whether relying on the mobile networks away from home or using my WLAN when I'm home.
I don't know what effect, if any, this would have on accessing the service via Roku, but Tunein has been known to roll out some bad updates to mobile OSs which are usually fixed pretty quickly and rolled out again. Apart from Tunein itself, I've noticed that some stations hardly ever work well no matter how good my local connection is, which I can't help thinking is due to output capacity issues on their end. As an example the main NPR outlet in L.A.--KCRW--is undoubtedly one of the their biggest, if not their biggest, affiliate in the country, but it's almost unlistenable on my Android device because of the constant dropping and buffering that occurs. When I want to listen to public radio I usually have to use the affiliate in Pasadena or San Francisco.