Forminator won't save form after "X" number of fields have been added

Once a form has several dozen fields, Forminator stalls when trying to save additional fields, and pops up a little red "Error! Form cannot be saved" message.

Also see a 404 error on admin-ajax.php in browser console when that happens.

Interestingly, when that happens, existing fields can still be edited, but no new ones can be added.