According to Facebook\'s Graph API documentation (here), you can access various sizes of a user\'s profile picture through URLs such as:
https://graph.facebo
I found a workaround - profile pictures of various sizes can still be accessed via an FQL query:
$pic = $facebook->api(array('method'=>'fql.query', 'query'=>"SELECT pic_big FROM user WHERE uid=$fb_uid"));
("pic_big" is equivalent to "type=large" - see here).
This still doesn't explain why the GRAPH call suddenly broke though, or why image sizes don't seem to be accessible via Graph at all anymore (which I'd still like to know)...but at least there's some way to get the other size photos.
Gotta love Facebook and their top-notch reliability...