jBackend does the following check to retrieve the api_key:

if (is_null($api_key))
{
// Get the input data as JSON
$json = new JInputJSON;
$json_data = json_decode($json->getRaw(), true);

The getRaw method of JInputJSON reads from the php://input stream (github.com/joomla/joomla-cms/blob/91749f845dcf20e097801ecd75da8e4665075a18/libraries/fof/input/jinput/json.php )

That means that for webservers that are on PHP < 5.6, the stream is now empty after reading the api key.
(See: php.net/manual/en/wrappers.php.php )

As such, all data is lost, and PUT methods cannot operate.

As a workaround, I am using a session variable to store the php://input data, by core hacking /components/com_jbackend/models/request.php

Because the jBackend uses the $json->getRaw() method earlier than my code (it uses it to check for existence of api_key and then passes control to my code), and thus empties the stream (remember this is for PHP < 5.6).