The Blame Game

I have fabulous tales of this weekend, but I need to get my Monday morning off my chest....

So I am going through Saturday's mail which had been stacked up somewhere to make room for the bodies in my house and I find a bill from our pediatrician's office. The bill is for $600 some odd dollars for Nora's most recent round of vaccinations. Odd, I thought those were covered.
So I call the billing number, they have not heard from Blue Cross about covering anything. Looking back, the woman found that every claim submitted for Nora had been denied, hence the exorbitant costs from when she was born. Weird.
So I call Blue Cross Blue Shield to see what the story is. They have no record of Eleanor being born or ever being covered by our insurance. What?! I was told to contact HR (Tim's mom) at his place of business and see if they added her to our policy, which it appears that they didn't. Oh, and by the way, we are still responsible for the cost of the vaccines because we missed the 30 day window to add her.
So I call Tim, who promptly hit the roof. He asked me to pull all the bills and stuff from her to see if she was once covered and when it renewed in August she was accidentally dropped. I am glad that I didn't get right on that because his mom just called me to let me know "what really happened".....
According to her it is all Tim's fault that she didn't add Eleanor to our policy in January. Sure she knew Nora's date of birth and stuff, but Tim "failed to remind her" to add the baby to our policy. He "should have known" that she was "out of it" due to "all that was going on in her life". Tell me, am I right to be annoyed that she is not taking responsibility for her own mistake? Or is it really Tim's fault for not telling her to add her to our policy?
How we ended things was by her telling me that she would "try to iron this out" by the end of the day, but if she couldn't get to it, it was going to have to wait until next Monday. She is really busy, you know. Meanwhile I am hermetically sealing Nora into some bubble wrap so that we have to pay no more out of pocket.
I hate HR.