Here's Wikipedia's take on this issue:
"Although a loose paper sheet is usually too soft to cut, it can be very thin (sometimes as thin as a razor edge), being then able to exert high levels of pressure, enough to cut the skin. Paper cuts are most often caused by paper sheets that are strongly fastened together (such as brand new paper out of a ream), because one single paper sheet might be dislocated from the rest. Thus all the other sheets are holding this dislocated sheet in position, and the very small part held away from the rest can be stiff enough to act as a razor."
(Paper cut)