I was thinking about this. A friend of mine was an engineer in a Cornish tin mine, and I'm pretty sure they had different levels. They had a system like twin oscillating ladders; one went up as the other went down, and you descended or ascended by stepping between the two moving ladders. When you reached the level you were to work on, you simply stepped off into the shaft. Each "ladder" had a central 'pole', with foot rests either side of it.
I could no more have done that than flown!! Imagine going down a 2 or 3000 foot vertical shaft in that way! (Shudder...) Yet, seemingly, they had few accidents.
Coal mines could be worked at different levels, providing each level was shored properly. Easier with tin mines, because the rock was granite, so it didn't suffer from subsidence in the same way; some coal mines definitely did suffer badly from subsidence, (as did salt mines) so it's doubtful if those were multi-level; but I can't see any reason why mines in good terrain shouldn't have been.
If you look at this article:
http://en.wikipedia.org/wiki/Longwall_mining you can see that multi-level mining was/is done in Western Australia, even though subsidence happens. What happens is that each seam or level is cut, the strata above allowed to fall after all the seam is removed, then coal strata above or below can be extracted as a new exercise. There is more subsidence with longwall mining (especially suited to coal) than with the pillar and gap method, but much more coal is extracted with longwall mining.
Either way, you wouldn't get me down there!!