Though this thread is tagged "Kindle 4", so it's probably not about Chrome on the desktop or some browser on the Fires, I guess.
My bet would be on SSL certificate or mechanism trouble, too. Either the cert store is (too) outdated - which I can easily imagine - or it is about SSL/TLS (though the K4 is new enough to use TLS 1.0), or encryption (though here, too, I think the K4 should support AES - as opposed to 3DES and RC4 - and SHA256 - as opposed to MD5 and SHA1). OP, test here:
https://www.ssllabs.com/ssltest/viewMyClient.html