Is it possible to tell if a user is logged into WordPress from looking at the cookies which are set?

You could use Javascript

<script type="text/javascript">
function getCookie(c_name)
{
var i,x,y,ARRcookies=document.cookie.split(";");
for (i=0;i<ARRcookies.length;i++)
{
  x=ARRcookies[i].substr(0,ARRcookies[i].indexOf("="));
  y=ARRcookies[i].substr(ARRcookies[i].indexOf("=")+1);
  x=x.replace(/^\s+|\s+$/g,"");
  if (x==c_name)
    {
    return unescape(y);
    }
  }
}

var logged_in=getCookie("wordpress_logged_in_[HASH]");
  if (logged_in!=null && logged_in!="")
  {
  alert("You are logged in!");
  }
else
  {
   alert("You are logged out!");
  }
</script>

NOTE: WordPress logged in cookie info can be found here. You’ll need to figure out what your hash is and possibly use regex for the hash. (this is untested but should work)

JS cookie source