Help Please — Cannot Log Out

Really weird. I cannot log out of WordPress. When I click logoout, it redirects me to: http://subdomain.domain.com/wp-login.php?loggedout=true and gives me the message “You are now logged out.”

…but as soon as I go to the page again, it has me as logged in! I’ve tried everything I could think of…clearing my cache, deleting cookies, etc. Am I missing something? Has anyone heard of this before?

Thanks!