My Chromecast has regularly been refusing to play Netflix streams recently with error ui-200.
Ignoring the onscreen suggestion and initial Netflix support page, a quick search will teach you to factory reset the Chromecast.
It doesn't answer the "Why?" though... It turns out that hitting the "Sign out of all devices" button triggers the issue. I guess the Chromecast stores a token which isn't invalidated or replaced, even if you log in again through the Android app.
Until Netflix/Google fix the bug, it might be time to think about upgrading your Netflix plan or telling "someone" to get their own account!