What causes 'Invalid form POST data' when removing an attached file from a node?
I can confirm, increasing 'max_allowed_packet' in our MySQL config solved this problem. MySQL's default of 1 megabyte was too small. Increasing it to 16M, then restarting MySQL has resolved this issue.

Info on our solution was found here:
https://www.shift8web.ca/blog/2014/11/drupal-problem-an-ajax-http-request-terminated-abnormally/