Should we be worried about these? We have some brown spots like these in the top floor, in the wall and some spots in the ceiling. Mostly around the ceiling nails like in the first picture. They are pretty small, but still. I tried to clean these kind of spots and they don't disappear, however they get lighter. We live in Northern Europe so we get some pretty harsh temperatures and climates, and it's a very old house.
It’s tannin oil bleeding out through the paint.
The first picture has a round (oval) shape that would normally indicate rust from a nail head. However, there is no blistering, so it can’t be rust. Tannin often will bleed out from knots.
The second picture shows the brown-ish stain in a long linear pattern. This is typical of tannin seeping out of siding (especially Cedar and Redwood siding).
All paint manufacturers recommend cedar and redwood should be primed prior to painting. Some paint types cover / block tannin better than others...oil based primers are best.
I'd hypothesize the nails extend into the wall where hot and cold air is mixing and causing condensation to make its way along the nail, through the wood, to the living space side of the wall.
The second pic would indicate both condensation within the wall and seasonal shrinkage/expansion of the wood to create the crack through which moisture can cause the paint to lift off the board at the joint.
Given this is an old house, I'd guess something was fairly recently done that decreases air flow within the wall (ie. exterior paint, new siding, caulking around windows and electrical outlets). Another possibility is that whatever moisture retarder there is (if any) has become degraded and no longer functions sufficiently. One final thought, check the roof, any gutters/fascia, and attic as increased moisture might be entering the walls from these areas.