Some solutions for $n=3$ can be found at http://www.research.att.com/njas/sequences/A010330 where there is also a reference to J. Leech, Some solutions of Diophantine equations, Proc. Camb. Phil. Soc., 53 (1957), 778-780, MR 19, 837f (but from the review it seems that paper deals with ${x\choose n}+{y\choose n}={z\choose n}+{w\choose n}$).
There are some other solutions at http://www.numericana.com/fame/apery.htm