You may receive the following error when you try to connect your WordPress JetPack plugin to WordPress.com:
Jetpack could not contact WordPress.com: register_http_request_failed. This usually means something is incorrectly configured on your web host.
Could not open handle for fopen() to https://jetpack.wordpress.com/jetpack.register/1/
There is not single cause for this problem. The above error occurs due to following:
 Your website is unable to call to WordPress.com servers to authorize JetPack. The possible reasons may be the following:
– WordPress.com servers’ IP addresses are blocked in your network or at network border
– Connection problem between WordPress.com servers and your website server.
– WordPress.com IP addresses are blocked in the server firewall
 Another cause could be mis-configuration of the server. If PHP and Apache are not configured properly, you may face this problem.
 WordPress.com is performing maintenance from their end.
There may be other causes for this error but so far I found the above causes for this problem.