When the console is loading, it appear to perform an operation which requires the controller lock (I'm not sure what yet). If you try to load the console while the controller has an in-progress operation holding the lock (e.g. stuck deployment in a domain) it will not finish loading.

The CLI lets you connect, and use operations to determine/cancel the long-running operation:
/host=master/core-service=management/service=management-operations:find-non-progressing-operations
/host=master/core-service=management/service=management-operations:cancel-non-progressing-operation

It would be good if the web console detected that it had connected to a controller but it was not responding and displayed a message. Using the :find-non-progressing-operations operation (which doesn't require the lock) to display what is stuck would be even better