All our infrastructure runs on Docker, and we do a .. lot.. of traffic with the customers we have now.
To be honest, Docker has historically been a pretty bad citizen in terms of having lots of hideous bugs that cause real production problems. Ask @mpalmer about that, he has the scars. But the good news is in 1.12.5+ and onward, Docker has gotten religion about fixing their stuff before moving forward with brave new world features.
(and with some prodding from us, I might add…)