Before getting Nick and Amelia, it had been many many years since I had a pet that required daily care and interaction. (My tortoise only eats a couple times a week, and she's happiest when I'm not messing with her, so she's very low maintence.) Going into rabbits, I hada LOT of anxiety about not being able to take care of them, and finding time in my life in them, etc.
For the first year, I had a lot of dreams about my babies, most of them bad. There was a dream where a stranger was trying to get to Amelia, and Nick tried to protect her but was kicked by the stranger. More than one where Nick was bitten and/or eaten by a snake. Many more, and they generally had bad endings for Nick, especially.
I believe these were related to the anxieties I had. I am now much more comfortable with my abilities to at the very least keep them fed and watered and supplied with toys, and I haven't dreamt of them in a long time. I would be happy to have good dreams about them, but at least I don't have the bad dreams anymore.