Picard: Warp 11, ensign. Engage.
Ensign: FFFFUUUUU...
It's been a while since I've seen any TNG/read the technical manuals/discussed this with one particularly fine example of the Trekker oevre... They were on the revised "Warp 10 equals infinite speed" scale, weren't they? (Whereas TOS, especially in STIV:tVH still had a continual logarithmic thing...)
Also, I'm pretty much assuming that they've ruled out that they're
overestimating the speed of the "we've sent a neutrino your way!" messaging system, right? While compensating for slow-down in the medium carrying the signal, gravitationally-induced relativistic effects, overlaying the effects of spinward/counter-spinward direction of the Earth, even buffering/latching behaviour in the microelectronics of the transceivers involved meaning it's the
second (or later) electron/photon of the signal that heralds the message, etc, etc. (And if they're synchronising with GPS clocks, and accounting for the mutual delays inherent in that system, accounting for certain atmospheric differences perhaps from the differing altitudes. And if they moved an atomic clock from one site to the other, account for the accelerating/decelerating frame of reference as well as all the previous.) They probably did, but there are a
load of things that need checking, many of which I know I've forgotten to even think of, even while dragging in some pretty inconsequential other examples.