If you haven't checked out my post about the leaked E3 games list you should go on and check it out, this news post will be about the Battlefield rumor surrounding it.
As you can tell by the image it clearly states Battlefield: Bad Company 3 as a title from EA to be shown at E3, now using common sense this is most likely a false rumor. Bad Company 2 came out in early March so if Bad Company 3 is going to be shown at E3 that is only 3 and a half months after the game came out to show the new one. Highly unlikely, it would destroy there DLC plans and most likely sales, who would want to buy things for a game that will be obsolete in the near future?
Furthering my belief about the falseness of this is that Barrie Tingle Producer from DICE replied to the rumor saying it is just that a rumor. The big thing there though he does not confirm nor deny the rumor just saying that it is infact a rumor.
"As I posted on the console forum... This is a rumour about BFBC3 being at E3."
"This is a total rumour about BFBC3 being at E3."
Now my next theory about this whole ordeal is that it may infact be a typo, and what is meant to be there instead of "Battlefield: Bad Company 3" is "Battlefield 3". I see that happening way more than I do Battlefield Bad Company 3, Battlefield 3 has been in developement for a long time, and I could definately see us hearing something about it this year, if it is at E3 or somewhere else.
Just because this rumor may be false does not mean we won't see something new from DICE at E3, Battlefield 1943 was very successfull. With that said I hope they pursue more games like Battlefield 1943, small downloadable versions of Battlefield. We do know they will be at E3 for Medal of Honor since they are doing the Multiplayer side of the game.