Just closing your browser isn't enough. Late one night I was doing financial stuff online. Merely shut down the browser after each site, as I always have done. Some time later I needed to go back to one CU. I launched the browser, went to the CU's homepage, and found to my alarm that I was put back into my account without entering my username and password. The previous online session was still open. Closed the browser and again got through without logging in.
It wasn't due to cookies being saved in my browser. I have my copy of Chrome with "Block third-party cookies and site data" and "Keep local data only until you quit your browser" both checked. And, of course, "Offer to save your web passwords" and "Enable Autofill" are both unchecked.
Tried the next day and didn't see the issue. The CU's site was probably backing up or being worked on by their IT people in the early hours of the morning. Somehow, the site's process that automatically logs off inactive users was not functioning.
I've changed my habits and now explicitly log off from any financial site using the button (usually located in the upper right-hand corner of every page on a site.) Just closing your browser isn't enough.