Crash bugs and weird glitches, sure. That can be on EA for forcing the game to market early. Fundamental shit like broken netcode and suspect hit detection, or shitty matchmaking? That's all on DICE. They've had many, many games to get that kind of shit sorted out.
The distinction is meaningless imo. Because ultimately they are in bed with each other since the buyout. You can't distinguish one from the other especially since no one here is there to diagnosis the relationship. And as people have mentioned, the regular consumer shouldn't care at all. He has no idea what the difference is between a studio and a publisher.
All the DICE shooter games have been buggy to very large degrees. They have a culture of this. Battlefield 4 is just the case where everything fell apart. And DICE is the one who has to fix the issue. They are the programmer's, coders, and artists behind the project. They are the only ones that can.
It also doesn't help when DICE isn't communicative to the public about the large scope of the problems. Whether that's PR, or executive talk, its frustrating to regular people when no one will acknowledge the scale of the problems.
This is a satire video of Patrick Bach a DICE higher up talking about how much better the netcode of Battlefield 4 was going to be than BF 3. He said a whole bunch of stuff and then tossed in this gem. Stuff like that pisses people off because they feel like they were lied too.
This was another pre-lease comment from one of the producers before the release of the game.
In 6 months #BF4 is going to be not only one of the biggest but THE biggest FPS Competitive game to play.
The competitive scene for the game died almost immediately because the netcode was so bad and the game stability was so poor.
So its one thing that DICE has been poorly communicating. That's unfortunate and laughably annoying in most cases. But the actual bugginess and actual problems with the game. Both in some poor design decisions that should be changed and the basic unstable nature of the game on many aspects is the real problem. No one will know what the real problem in the relationship is. There are clearly problems on both sides. But as a consumer I want the product to work. So DICE/EA share the credit when things go well like in BF 3. And they both get the blame when things turn to shit. Like in BF4.
Eventually when BF 5 rolls around you will hear the mea culpas about how flawed BF 4 was to sell people on the idea that they've learned from the huge mistakes in BF 4 as they want you to buy the new edition. It's just unfortunate a lot of people had to suffer through a janky product and spend $100 bucks on it without feeling good about that product.