I had this issue when attempting to save a configurable product, and it was due to the sheer amount of individual configurations assigned to the parent product (over 1,600!). Whenever you save a configurable product, it also processes each of these simple products, leading to one large Ajax request (at least 1MB). When I analysed the network information within the browser's Developer Tools, this Ajax call was throwing a 413 "Request Entity Too Large" error.
The solution for me was to adjust the settings on your server to handle larger requests. If you're running Apache, you can add
LimitRequestBody 8388608 to the .htaccess file (I think); if you're running Nginx, add
client_max_body_size 8M; within the http, server, or location context within the nginx.conf, and then restart the service. These will both increase the maximum size to 8MB.