Are you running Origin (and Mass Effect 3) as an Administrator?
1) Right-click the Origin icon on your desktop and select Run As Administrator.
2) Restart Origin and try launching your game.
Do you have any busy Virus scanners, or anti-malware programs running?
Try installing again with Task Manager running and watch it to see if any other processes or 3rd party applications appear to be doing something funny.
The error sounds like you cannot communicate with the server hosting the software. Do you have any other network issues? Multiplayer works fine?
How direct is your computer communicating to the internet?
I assume PC using LAN (Wired) -> Router using LAN -> Modem.
Perhaps somebody at Origin can tell you what server or IP address to ping to be certain.
If you have two routers or some other device added in the mix, it may cause problems. (I've had to replace switches before)
Conspiracy theorists might have something to say about Net Neutrality, if your ISP is funny.
