Seems like you'd be punishing users instead of website operators since the cause/effect relationship is so unobvious.

User happens to brush over the logout button while using the site. On their next click, they're logged out. Weird. Guess I'll just log in again. Doesn't happen again for some time, but then it does. Weird, didn't that happen the other week? What's wrong with my browser? Oh cool, switching browsers fixed it. You're having that issue, too? Don't worry, I figured it out. Just switch browsers.