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"POSIX error: Operation Timed Out": Make sure your device has access to the internet and try again. If this error keeps happening, it is likely caused by a repository server being temporarily offline. Try again later. You can also choose to remove the repository.
The only thing that I have found to work is completely rejailbreaking the device.
The problem is not temporary and despite what the message says, it seems to have nothing to do with a server in any way. It is possible for a device to get this message, and will continue to repeat this message indefinitely every time Cydia is launched. I don't know why. Once I rejailbroke my iPad the problem instantly disappeared.