] She may, for example, have spent her whole life being told she was too ill to do much, and after a while start to believe it.

I can see this as a likely scenario. To add to it, I can also see LC, after having subconsciously urged Anne to behave as an invalid, being frustrated with her for being so. She seems an active, energetic woman herself - as indulgent as she is to Anne, I don't imagine her coddling the "illnesses."

The dialogues in my head:

LC: Oh, Anne, you look as though you are feeling ill again. I believe we must postpone our trip to London for your preesentation yet again.
A: Well, now that you mention it, Mama, I do have a bit of a pain in my neck...
LC: Well, then, that's settled. I am excessively attentive to the health of my daughter!
(a few hours later)
LC: Anne! Are you merely going to sit around the drawing room all day?!
A: But I'm ill, Mama, you said so.
LC: Well, sitting within doors will do you no good. Get out into the fresh air. A little activity and movement will put the color back into your cheeks! Mrs. Jenkins! Order Miss DeBourgh's phaeton at once!