Silly me, of course the answer was not number of fields, but number of post vars…
72 fields just happened to push the post vars count over PHP 5’s default limit of 1000. This issue also crops up with really large menus in WP.
The fix:
Add a php.ini file to /wp-admin/ and put these settings in there (assuming your host allows php.ini overrides):
max_input_vars = 3000
suhosin.post.max_vars = 3000
suhosin.request.max_vars = 3000
From: http://anothersysadmin.wordpress.com/2012/02/16/php-5-3-max_input_vars-and-big-forms/
