Now this here's a story all about how I got absolutely shoved to the side by EA support AND MSFT support...
I have been having the same EXACT problem as everyone else here for about the entirety of the month of may so far. It began with what you describe and I performed the numerous possible fixes for the issue supplied on the interwebz. To no avail, I called the EA big boys. They sent me round and round with this dude on support for about an hour and a half and I finally got bumped up to the advanced level support or something of the like. This british guy with a glorious voice told me that the BF servers all around were experiencing some hard hits on the Battlefield servers. I was told by this fella that it is most likely this attack causing the issue and that my login error was due to the entire conglomerate of Battlefield servers being hit all at once, including the login servers which handle player connections exclusively.
WELLLLLL...
The following day, I tried logging in during the afternoon and encountered the same error. Mind you I supplied the new EA support tech with the case number from the previous daunting call and was instantly forwarded to an advanced support rep. This rep spent about 2 full hours on the phone with me going through TONS (and I mean TONS) of possible fixes to the issue.
The list of possible fixes are as follows:
-Resetting your email to match your email associated with your gamertag
-Taking away permissions from any other gamertag associated with the EA account EXCEPT for the gamertag in question.*
-Reverting the email to the original email used to make the EA account if it was changed post buying the game.
-Entering in a new redemption code for the game TWO DIFFERENT TIMES with TWO DIFFERENT CODES to see if that re-enabled access to the game to my account through EA Online and/or XBLA.
-Trying to reset my password through another EA Xbox Live app**
-Try logging into the Battlefield Bad Company demo for Xbox (didn't work because demo servers are shut down)
Fixes that I was not able to perform because I didn't have certain disc-based EA online games:
-Try logging into Battlefield 3 servers through Xbox
-Try logging into Mass Effect 3's multiplayer servers through Xbox***
-Try logging into Battlefield 2 Modern Combat through Xbox
All in all, once I was done with the support rep, they told me I had to have my case escalated to the DICE development team considering they haven't seen this issue before, ever, at all, in history. Wonderful lucky me.
Aaaaaaaand to this day I still have yet to receive any sort of email from either EA support or DICE regarding the issue. So I'm essentially screwed out of a game I've loved since launch because of an unsolvable network issue specifically pertaining to my gamertag.
*=(This idea came from the fact that I could play 1943 IN FULL on my secondary XBL account that has never even purchased the game. It was only able to do this AFTER I signed out of my main XBL account and signed into my secondary WHILE STILL in the game's menu)
**=(didn't work simply because I couldn't reset my password through any other EA online enabled app on Xbox Live through my Gamertag)
***=(Though I was able to log into other games such as the FUSE demo and FIFA 13 demo and was able to play online)
P.S. I say MSFT support as well because when I called about the issue, they helped up until the point I mentioned that I was told DICE was being contacted by EA regarding my issue and I was told that no other fixes can be applied considering the measures taken by EA support and that they can't do anything until EA follows up.
At least EA guys were persistent as hell and helped until the last breath.
P.P.S.
1. ADAM Addict13
2. Comcast
3. Florida USA
4. 3+ weeks