I think they mean that they cannot log in... When they're with us they log in as "CPDLC" and when for some reason they cannot (something broken up there or down here) that's when they say "negative CPDLC".
Not that we wouldn't see it, but I guess it's just the way it's taught at BAW... The other companies don't use this phraseology that much.
Wait for a Japanair - when they have it they say "we have CPDLC connectioooon"