Constantly dropping and/or stalled connection between worker role & Azure VM

For more than 8 hours we are experiencing connectivity issues between Worker Role (.NET bundle) & Azure VM (RabbitMQ server on Windows Server). The code was not changed - it was working without any issues like that for more than 1 month (even yesterday during the highload period of our system everything was OK).

We've got the following issues

1. Failed to receive confirmations from Rabbitmq even for the smallest messages (WaitForConfirmsOrDie exceptions)
2. During the fault periods - Azure VM is unaccessable via RDP & Rabbitmq management port
3. We are not seeing any high CPU usage on both VM & Cloud service
4. We've restarted Azure VM & we have lost RDP connectivity to it (connection drops after password & certificate verify dialogs)

We had to redeploy the Stage environment for several times - and each try was stable for only 2-3 minutes. We can see the issues with connectivity even from the external network (using our office&home PCs).