I would be sold on Docker if that would be easy. I have e.g. this stack: - 1 webserver/proxy, let's say nginx - 1 simple Rest API server, let's say in flask - 1 database, let's say PostgreSQL and I want to connect all 3 things and I want to preserve logs for the whole time and preserve the state of the database (of course). Also not to forget make all bulletproof for the Internet. And here all sorts of problems arise…
Databases are also tricky to run in containers, because even those with the best replication strategies can afford losing nodes but at a high cost (like re-balancing nodes, etc), and containers still don't have the stability to provide an acceptable uptime that's worth the risk.
On a side note, since you mentioned nginx and RESTful APIs, I would check out Kong (https://github.com/Mashape/kong) which is built on top of nginx, and provides plugins to alleviate some of these problems (http://getkong.org/plugins/).