Mixture, the banking site lets me into the main website as you say, but then displays the can't display page message when I try to log in. My main email site, (hotmail), won't even let me access the main page - it displays message as soon as I type the address in the browser & hit the enter key. Confusingly, it has no problems letting me in to my gmail account (only used as secondary email).
Assuming its the browser displaying the page (i.e. the message is the same, not a message thrown up by the site), I suspect what you might be seeing is your present browser has a cryptographic library incompatibility that means it cannot talk the specific ciphers that the website is willing to talk. So the browser throws its proverbial arms up in the air and gives up.
As a not too technical user, I've never actually upgraded my browser before, so should be interesting....
I hope you've at least been running SOME software updates on your machines ?
