Not since they did the redesign on the site a year or two ago, before then I could. I agree its annoying all right but at least autocomplete now remembers the text of my user name in the form field which it did not do after the redesign either until recently.
Sorry - I did a bit more testing and now see that once the session is closed (e.g. the tab in FireFox or the browser is closed) then the login session is terminated. This is because the site uses session cookies to track the login as far as I can see. The login form does remember my email/password though so I can just click login when I return to the site but am logged out.
Are you using Firefox? If you inadvertently clicked on the "never for this site" button the first time you put in the password, it will never allow your password to be saved once the session is over.
There is a way to reset this, of course if you're not using FF it's completely academic!
HTH