When you tick the checkbox, the site sends Firefox a persistent cookie so that even after you exit and start Firefox up again, Firefox can send the cookie back to the site. (Otherwise, you get a cookie that's only valid for a short time.)

There are many possible reasons that a cookie might not survive long enough to get you back in the site. These include changes to Firefox's cookie settings, or using a private window, or add-ons that clean up cookies, or external privacy/utility software that cleans up cookies.

Can you test how soon Firefox forgets? For example, if you log out of your bank and then open a new window and go to the login page again, if your user name there or blank? A second test would be exiting Firefox and starting it back up again right away. (Sometimes external cookie cleaners only run once a day at a set time, or run when you start up or shut down your computer.)

Clear the cache and remove cookies only from websites that cause problems.

"Clear the Cache":

• Firefox > Preferences > Advanced > Network > Cached Web Content: "Clear Now"

"Remove Cookies" from sites causing problems:

• Firefox > Preferences > Privacy > "Use custom settings for history" > Cookies: "Show Cookies"

If clearing cookies didn't help then it is possible that the cookies.sqlite file that stores the cookies got corrupted.

• Rename (or delete) cookies.sqlite (cookies.sqlite.old) and delete other present cookies files like cookies.sqlite-journal in the Firefox profile folder in case the file cookies.sqlite got corrupted.

• http://kb.mozillazine.org/Cookies
• https://support.mozilla.org/kb/Deleting+cookies