When JConsole is monitoring a local process and host's IP address changes on-the-fly (you can try changing it manually), JConsole loses connection to the process and unable to restore it anymore from the both 'Reconnect?' and Connect dialogs. Host name had also been changed in my case.

If I start another instance of JConsole, it is able to connect to itself, but is hanging up on connecting to the first JConsole (locally).

The same also happens when dial-up Internet modem line disconnects (a Cisco VPN client is installed on that dial-up machine. It may have an impact on TCP/IP stack behavior).

Observed on Windows XP. Not checked on Linux and Solaris yet. Also I'm not sure whether it JConsole or JMX bug.
Same happens on Vista after changing IP address and clearing DNS server fields.