Please check your logs for errors, you should have clues there.
Also check if your disk didn't fill up. I would guess your problem is caused by PHP not being able to write session files to disk. This could be a full disk, but it could also be permissions problems.
In your php.ini there is probably a variable called session.save_path which tells you the directory where it's trying to save. Have a look there (how many files are there? Is the web server able to write new files there=?)
Thank you for your reply. Before I could try your suggestions I was reading another thread on the same problem. It was suggested to check to see if there where multiple instances of SuiteCRM running. I have 2 identical physical servers setup. One is "live" and one for updates/testing/backup. Once in a while I switch the "live" SuiteCRM from one PC to the other. Somehow I got both running same "live" instance in the respective VMs, and connected to the network at the same time. 2 days of headaches of my own doing. As soon as I stopped one VM all our logout problems have magically disappeared.
The administrator has disabled public write access.