Wet hay fires still happen here once in a while, although wrapped bales stored outside are more common now and less likely to burn.
According to this the temperature required is far less than the 450 F ish that it takes for paper to burn.
apparently it's the accumulation of volatile gasses that burn so the temps are lower.
http://ext.wsu.edu/hay-combustion.html
For the most part though, good hygiene and the elimination of smoking has done a lot.
It astounds and ticks me off how careless people are with fire.
It's entirely possible for birds to bring a discarded lit butt back to the nest, which is a perfect tinder bundle.
I think the safety codes that specify can rags also include a lid, usually a self closing lid.
I assume the lack of oxygen is what reduces that risk.
I tend to use a rag once, so no accumulation of several rags.
After using linseed oil rags I've always either burned it right away or tossed into a firepit (safe if it did burn) or, rinsed in water and hung to dry outside, or put into a sealed can.
Heat, Fuel, Oxygen try and remove 1 or 2 of those and you won't have fire.
I almost burned the shop down with sparks on an oily rag too.
It was smouldering and I couldn't smell it with the respirator on.
I think it's good practice to have 1/2 hour clean-up / down time after working to prevent or detect this sort of thing.