Hi Sandy and welcome. I have seen this problem, you're not alone. Would you by any chance be using IE on a Macintosh?
A few weeks ago I was on holiday, and at one point I managed to scam some time at a friend's house who has a Mac and IE installed. I noticed that there was no way to get IE to save the cookie. Each time I tried to log in to PM, it would keep coming back as having me logged out.
I finally found that if I just logged in without asking for a cookie (don't check the checkbox next to the password field), it worked for as long as I kept the browser window (read: session) open, which, when it comes down to it, was just what I wanted, given the circumstances.
I have no idea whether it is a bug or a (security(?)) feature of IE interacting weirdly with PM, but there you have it. Does this sound like your environment? Maybe it's time to install the lizard. At least just trying a different browser might help determine where the problem lies.