SOLUTION:
I had the exact same problem, but this worked for me. Both singleplayer and multiplayer.
Go to http://par.nofate.me/ .
Submit the "BF1942.par" file. You'll find it in the Origin games folder. The icon for the file is a white sheet of paper.
After submitting, your computer will download a new file with the same name. Delete the old "BF1942.par" file and replace it with the new one.
Try starting the game. NOT IN ORIGIN. But outside of Origin. It'll ask for your username and password. And voila, it should work.
