This is an extract from Apéry's biography (which some of the people have already enjoyed in this answer).

During a mathematician's dinner in Kingston, Canada, in 1979, the conversation turned to Fermat's last theorem, and Enrico Bombieri proposed a problem: to show that the equation $$ \binom xn+\binom yn=\binom zn \qquad\text{where}\quad n\ge 3 $$ has no nontrivial solution. Apéry left the table and came back at breakfast with the solution $n = 3$, $x = 10$, $y = > 16$, $z = 17$. Bombieri replied stiffly, "I said nontrivial."

What is the state of art for the equation above? Was it seriously studied?

Edit. I owe the following official name of the problem to Gerry, as well as Alf van der Poorten's (different!) point of view on this story and some useful links on the problem (see Gerry's comments and response). The name is Bombieri's Napkin Problem. As the OEIS link suggests, Bombieri said that

"the equation $\binom xn+\binom yn=\binom zn$ has no trivial solutions for $n\ge 3$"

(the joke being that he said "trivial" rather than "nontrivial"!).

As Gerry indicates in his comments, the special case $n=3$ has a long history started from the 1915 paper [Bökle, Z. Math. Naturwiss. Unterricht 46 (1915), 160]; this is reflected in [A. Bremner, Duke Math. J. 44 (1977) 757--765]. A related link is [F. Beukers, Fifth Conference of the Canadian Number Theory Association, 25--33] for which I could not find an MR link. Leech's paper indicates the particular solution $$ \binom{132}{4}+\binom{190}{4}=\binom{200}{4} $$ and the trivial infinite family $$ \binom{2n-1}n+\binom{2n-1}n=\binom{2n}n. $$