I have exactly the same issue.. Using OS X 10.10.5 on a Macbook Pro, 17", mid-2010.
Facetime worked at some point, but now it just refuses to log in. It either sits there until I hit cancel, or after a seemingly random amount of time it will say exactly what iJudaz reported above.
Could not sign into FaceTime.
Could not sign in. Please check your network connection and try again.
I do have LittleSnitch installed, but I've had that for years. I have no rules that block anything related to Facetime. What's more, even if I turn it off, the issue persists.
In terms of troubleshooting... I can see in the LittleSnitch network monitor that FT connects very briefly to profile.ess.apple.com when I first start it, although at that point I've not even entered my password. It transfers almost no data... 1.60 kB up, 21.7 kB down. Other than that, it makes no further connection attempts or data transfer.
If I hit CANCEL and then type in my password and click Sign In, NO data is transferred over the network at all. It sits there completely dead, as far as network connectivity is concerned.
This suggests to me it has a serious internal issue, as opposed to issues with the network connection as the error message suggests.