This is very odd.
I have one machine running Opera 10.53 that has been sequentially upgraded since 9.x. This latest upgrade has done something odd that I can't replicate on other machines. When I try to sign in to Twitter or to access Gmail's login page or several other HTTPS sites the browser just keeps trying to load the page, unsuccessfully. Last week I tried to log in to Twitter in one tab and 3 days later it was still trying.
The odd thing is that this is not consistent across other sites that use an SSL certificate. Some sites I can log into without a hitch. Other browsers like FF and Chrome have no problem with these same logins.
My first troubleshooting step was to uninstall all instances of Opera, reboot, and start afresh with a new copy of 10.53. That didn't change things. Any other steps you might suggest? I'm going to cull my cookies now and play with security settings...