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:
[1] 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
[2] Another cause could be mis-configuration of the server. If PHP and Apache are not configured properly, you may face this problem.
[3] 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.