Ann and Bob's cooperation is jointly necessary for doing something that both are morally obligated to do. Ann and Bob can't agree on how they should go about doing that thing. Ann refuses to do it Bob's way, and Bob refuses to do it Ann's way. In both cases they believe they are morally required not to do it the other's way. So Ann sets out to do it her way, and Bob refuses to cooperate, because he believes her way is immoral. Ann then complains that Bob is refusing to fulfill his moral obligation. Bob complains that Ann is refusing to fulfill hers. The obligation does not get fulfilled.
Can Ann claim that Bob (and Bob alone) is refusing to carry out that responsibility? Can Bob say the same of Ann? My impression from the case as I just explained it is that neither is any more or less responsible than the other for not completing the obligation. Both are equally to blame, and both are somewhat to blame. But consider a slightly altered example. Ann and Bob can fulfill their moral obligation by cooperating, but it would mean Ann does not do something that she also thinks is morally required. Bob wouldn't be sacrificing any moral obligation he believes he has to fulfill the one, but he thinks he'd be doing something wrong to cooperate in the other.
In this case, Ann refuses to do it, because she thinks she ought to do both, and if Bob won't let her do both then she'll do neither. In this second case, then, Ann is morally to blame for not doing the obligation that both agree they have, and Bob is not to blame for not fulfilling that obligation. The fact that Ann thinks there's a further obligation that Bob doesn't think he has does not give her the moral freedom to abandon the one obligation she can fulfill, since it's better to fulfill one moral obligation that you can fulfill even if there's no way the other person will let you do what's necessary to meet the other obligation that you think you have.
Now consider the Congressional leadership and President Bush on the issue of funding of troops in Iraq. Both parties agree that they have an obligation to fund the troops in Iraq right now. The Congressional leadership thinks they have a further obligation to get the troops out of there very soon with an explicit deadline. Bush disagrees. He in fact thinks he has an obligation not to allow that. He thinks they have no such obligation. Now they fashion a method of doing both at once, but he considers that meeting one obligation (temporarily) while violating another. If they followed his recommendation, they would be meeting one obligation while not meeting what they consider to be another one.