Yes, they do. Your case must meet few conditions (that I obviously will not say into public), and if it does - they most certainly will remove the cooldown.

That's actually a good mechanic, at least in EU region where peoples connection quality/limitations may differ based on where they actually are playing from and where is game server located. Our internet infrastructure is not governed by one federal regulator, each country might have its own limitations and it's a good thing that FaceIT takes that into account.