My tiel Shriek had night frights just a couple of times, and never during a storm. (I think it's dreams, something you can't control.) He was always covered at night, and like the others who get in the routine, would let me know if I was late. My Jardine's came to me never having been covered, but it didn't take long to get her used to it. Our house is small, and the living room lights illuminate her cage very well. Since I'm trying hard to keep her on a strict day/night schedule, it seems to me nighttime should be nighttime.
The funny thing is that I'm using the same little blanket with her new (huge) cage. It only covers the top third of the cage at most. She's very comfy with this arrangement. But I may have to get a bigger blanket. A couple of weeks ago she got very active during some brightly moonlit nights.