saggy drywall - or some other panels they used and plastered over?

Parts of it look like a bad skim coat. But the rest does look a bit saggy. If you push up halfway between creases, do you feel any "give"?

Maybe it was done with the early style of "drywall", which was squares of something, not big sheets.

We have similar lines in our ceiling and its because we have plaster over rock-lathe. Rock-lathe was often made as 4'x4' sheets of material similar to drywall, but with a very moisture- permeable coating (and sometimes small holes) that allowed the additional plaster coats to stick to it.

When you see seams like that it usually means the nails holding the rock-lathe have started to pop. The plaster layer is still holding everything together, so its usually not going to fall down immediately; however, it may eventually sag enough to pull the plaster down. If a sheet of that plaster drops it will be very heavy, and it will damage anything that is under it.

Assuming you don't want to tear out the plaster, the simplest repair is to have someone carefully jack up the ceiling, and screw 1/4" drywall on top of the plaster. They need to make sure they use long screws and pull both the drywall and plaster up tight against the joists. Then just finish the drywall as usual and paint the ceiling.

Bruce

You better c-a-l-l somebody.

Also, check and make sure there is no roof leak or excessive moisture in the space above.