Normally the backup server would take over, but it isn't; only the documentary server is available.
This comes back to the good 'ol Comms Rule: "If the OS can't see it, then neither can Notes/Domino." Given that a couple of your machines are not having any issues I would suggest that your problems are network-based, rather than Notes client-related, especially since you have indicated that the Notes clients are able to show "connected" messages in the status bar, and your replicator (dialogs/pages) are not showing "Unable to find path to server: Failed with..." errors.
However, you have suggested that you can "ping" your destination, and that suggests there is SOME connectivity at an operating-system level. This might be related to the bandwidth and availability of the connection to your servers. Try changing the time-out settings to test this.
[File] -> [Preferences] -> [User Preferences], "Ports" tab, [TCPIP Options] buttons, change the time-out interval to something fairly high (e.g., 30-60 seconds) and see if that makes any difference.
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