Initially I was thinking Munchausen as we had a lady at work who was tagged with Munchausen by proxy. She kept causing her daughter to become ill, then take her to the emergency room. This woman was otherwise smart and likeable, but she lost her child due to the syndrome.
I do know a few friends who behave as you describe and it simultaneously makes me angry and breaks my heart. You just want to shake them to make them realize, but as you say, you can't make them see any different.
Fortunately, I've only encountered Munchausen by proxy in the movies. Those people are really messed up...
Like most stuff in the movies, most of these accusations are false. If anyone has Munchhausen's by proxy, it will be the doctor entrusted with the child's "care." They love to blame the mother when their remedies fail.
That is the new diagnosis of choice. Doctors love to diagnose people with that, but they are almost always wrong. Any mother who knows a lot about their child's illness, and what mother would not want to know a lot, are diagnosed with that nonsense regularly. I have been accused of the same. I feel sorry for her, and especially for the child, who is probably is much more sick now than she was under her loving mother's care.