I believe the first answer to your question is the right one. You've ruled out some kind of timed filter, but the gist is that the issue is probably your connection. I believe the actual problem is that your ISP's connection to the hub is flaking out.
One test you can do is running the tracert tool on hub.byond.com during a time nothing is working. If the trace stops at a particular point upstream, or starts bouncing around the same servers over and over, you have your answer.
The message "Another connection has been made using your login key. This session has been terminated." message shouldn't necessarily be cause for alarm and is normal. Dream seeker displays this message and action anytime that you make a connection to a game that dream seeker is already connected to/trying to connect to. If you tried (accidental or otherwise) to launch two instances of a game, it may have resulted in that message.
However, if you do feel your account is in jeopardy, then , as suggested, you should change your password and email address.
Although the above is a separate issue and shouldn't be confused with your initial issue (and shouldn't be reason for immediate concern).
Regarding the initial issue:
The change in IP address at the time the problem start occurring may be what's going wrong, although I am not sure why. Please try these tests.
1) When the problem doesn't occur, does your IP address revert back to the original IP address as when the problem didn't occur the previous day?
2) Does your local IP address remain the same the entire time? (see post#3 for instruction on how to retrieve your local IP address).
3) Have you followed the steps in post#5? (If you are unsure where the BYOND folder is, Open the BYOND pager > File > Preferences > Advanced > Open User Directory) Did they have any change in results?
4) When the problem occurs, are you able to join games by logging into the BYOND pager as a guest?
5) When the problem occurs, try running a tracert on hub.byond.com (Start > Run > type cmd > enter > type tracert hub.byond.com > enter) to see if you are actually able to reach the hub.
6) When the issue occurs, Does disconnecting the power from your modem/router (for a minute or two) result in any change?
Good luck!