Update information Edit Edit2(Jun.22)
WordPress 3.9 named “Smith” is available but waiting Japanese version. How about you, guys?
Edit:
I upgraded to WordPress 3.9-ja at 20:19. I still have the same issue when I updated to 3.8.3. The issue is that Warning! Problem updating https://SITENAME. Your server may not be able to connect to sites running on it. Error message: error:14094410:SSL routines:SSL3_READ_BYTES:sslv3 alert handshake failure. Why does it happen???
Here are the information pages about WordPress 3.9. Codex, Blog, Changelog.
Edit2(Jun.22):
I wrote an article The solution of “SSL3_READ_BYTES:sslv3 alert handshake failure” on WordPress.