Please note: This post is several years old now and may no longer be valid. If it does not work for you please check your logs for the new solution. I would love to hear about it to update guidance on this post if needed.
If you are running into the following error:
Jetpack could not contact WordPress.com: register_http_request_failed
It is likely because your host does not properly support outgoing HTTPS requests. To get around this you can disable the Jetpack HTTPS call and run over HTTP by placing the following in your wp-config.php file:
define( 'JETPACK_CLIENT__HTTPS', 'NEVER' );
Note the double underscore between CLIENT and HTTPS.
I want to stress that though this will work all your interactions with the Jetpack servers are now no longer encrypted. If you are not concerned about the data being transmitted ( this could include comments ) it is ok to leave that line in wp-config.php but it is much safer for you and your visitors to run over HTTPS. I highly recommend contacting your hosting provider and asking them to enable outgoing HTTPS.