Page template dropdown missing when editing

So the weirdest thing – I have a custom theme, and I’m in the process of transferring from one server to another.
Everything is going well, all posts and pages showing properly, but the pages are no displaying properly, and the dropdown menu to select the page template has disappeared.
However, if I go to Quick Edit it is there.
But even if I choose a template in Quick Eit and update the page, the template is not used for the page.
I have tried everything I can think of –
– changed to twentyeleven and then back to the custom theme
– checked to make sure that I have the right code in the template’s php file:
<?php
/*
Template Name: custom
*/
?>

– deleted and reinstalled the custom theme
– reinstalled WordPress itself

I am at a loss to understand why the templates menu is not showing up.