A binomial generalization of the FLT: Bombieri's Napkin Problem

Some solutions for $n=3$ can be found at http://www.oeis.org/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

EDIT Here are some more references for $n=3$:

Andrzej Krawczyk, A certain property of pyramidal numbers, Prace Nauk. Inst. Mat. Fiz. Politechn. Wrocƚaw. Ser. Studia i Materiaƚy No. 3 Teoria grafow (1970), 43--44, MR 51 #3048.

The author proves that for any natural number $m$ there exist distinct natural numbers $x$ and $y$ such that $P_x+P_y=P_{y+m}$ where $P_n=n(n+1)(n+2)/6$. (J. S. Joel)

M. Wunderlich, Certain properties of pyramidal and figurate numbers, Math. Comp. 16 (1962) 482--486, MR 26 #6115.

The author gives a lot of solutions of $x^3+y^3+z^3=x+y+z$ (which is equivalent to the equation we want). In his review, S Chowla claims to have proved the existence of infinitely many non-trivial solutions.

W. Sierpiński, Sur un propriété des nombres tétraédraux, Elem. Math. 17 1962 29--30, MR 24 #A3118.

This contains a proof that there are infinitely many solutions with $n=3$.

A. Oppenheim, On the Diophantine equation $x^3+y^3+z^3=x+y+z$, Proc. Amer. Math. Soc. 17 1966 493--496, MR 32 #5590.

Hugh Maxwell Edgar, Some remarks on the Diophantine equation $x^3+y^3+z^3=x+y+z$, Proc. Amer. Math. Soc. 16 1965 148--153, MR 30 #1094.

A. Oppenheim, On the Diophantine equation $x^3+y^3-z^3=px+py-qz$, Univ. Beograd. Publ. Elektrotehn. Fak. Ser. Mat. Fiz. No. 230-241 1968 33--35, MR 39 #126.

My first instinct is to say it seems unlikely there's been serious progress on this problem for general n. Unlike the Fermat equation, this one is not homogeneous of degree n, which means that it's really a question about points on a surface rather than points on a curve. We don't have a giant toolbox for controlling rational or integral points on surfaces as we do for curves.

In fact, I can't think of any example of a family of surfaces of growing degree where we can prove a theorem like "there are no nontrivial solutions for n > N." OK, I guess one knows this about the symmetric squares of X_1(n) by Merel...

Another paper that mentions the problem is "Explicit Solutions of Pyramidal Diophantine Equations" by L.Bernstein Canad. Math. Bull. Vol. 15(2) from 1972! In fact I realized that this problem could have appeared in literature long before expressed in terms of "figurate numbers". Anyway an interesting list of references (I haven't found most of them yet though) can be found on section D8 of R.Guy's "Unsolved Problems in Number Theory".

Also two more OEIS links with useful information. I would also like to find this article by H. Harborth, "Fermat-like binomial equations", Applications of Fibonacci numbers, Proc. 2nd Int. Conf., San Jose/Ca., August 1986, 1-5 (1988). (Link)

As a conclusion, the problem has been mentioned in several papers, and many special cases have been given a lot of attention. Bombieri doesn't seem to be the original source of the question.