@lehren
Thanks for providing the link to the twitter post you’re trying to embed.
Since you mentioned the issue is happening on your site, I tried it on different websites hosted on different platforms and locations.
Apparently, on some of the sites its working fine, and on others its not.
I dig a bit deeper and looked for possible issues, turns out, Twitter API for oEmbed is returning “Method Not Allowed” (header 405) when requested from some of the servers while returning the correct response for others. So I’d say, nothing is wrong at the WordPress end, but its Twitter API that’s refusing to serve some servers, while allowing others.
To give you an idea about how oEmbed works, WordPress makes an API request to the service whose URL is going to be embedded, then based on response, the HTML markup is generated, but if those services do not respond or give any errors, WordPress cannot generate the HTML markup and gives you an error without stating what exactly happened.
Would be great if you can contact Twitter support for this, asking why your server’s IP is not receiving the responses while other IPs are.