Very rarely. 99% of the time when I sit down to watch a movie/video it's something I've downloaded and stored on one of my home servers. The result is that I'm streaming, at very high bitrates over a gigabit local network, a movie/video in full, near-lossless audio/video quality, with no need to buffer in the event of a slow Internet connection (which can happen from time to time if you rely on YouTube to watch most of your content directly). I can also still watch my content even if I lose Internet service completely. As for Chromecasting local files from my phone/tablet, I don't tend to download videos to these devices at all, that's all what my server is for. When it comes to videos I take myself on my phone, they sync to my server as soon as I walk in the door using BitTorrent Sync, so it's ready for viewing by the time I take my shoes off and hop on the couch. Plus since my PS3 would be on if watching videos anyways, I can just control the YouTube app using my phone if I did want to stream something from YouTube directly. Since I don't use any other streaming services, all other use cases for the Chromecast go out the window for me.
tl;dr - Any use case I would have for a Chromecast is already handled by something else that's better, provides higher quality, and takes marginally more time to prepare, which is worth the negligible tradeoff anyways.
The Chromecast is great for people who want a simpler set up that's quick and easy to get up and running, but isn't as arguably reliable and doesn't provide as high quality content. But it just works, and it's cheap, which is the point.