We’re hosted on WP Engine, who says this about max_input_vars:
Max Input Vars
The default max_input_vars setting is 10000, indicating no more than 10,000 variables can be attached to any request.
This setting cannot be adjusted, as this is set at a platform level and higher values will have negative performance implications.
Granted, 10,000 seems like it should be enogh. Hmm.