Are there surprisingly identical binomial coefficients? Suppose $\binom{n}{k}=\binom{n'}{k'}$ with $k \geq 2$, $k' \geq 2$, $n \geq 2k$ and $n' \geq 2k'$. Does it follow that $n=n'$ and $k=k'$? 
EDIT: Yup, $\binom{16}{2}=\binom{10}{3}=120$. 
Now I want to ask if there are infinitely many such pairs, but I should probably ask that in a separate question. Thanks!
EDIT 2: for future reference, yes there are infinitely many such coincidences. See Singmaster's Conjecture.
 A: $$
\binom{3003}{1} = \binom{78}{2} = \binom{15}{5} = \binom{14}{6}.
$$
Nobody knows whether any number besides 3003 appears at least eight times in Pascal's triangle.
A: Let $f(t)$ be the number of ways in which $t$ can be written as a binomial coefficient. Then the question can be restated equivalently as asking whether $f(t) \leqslant 2$ for all $t$. As the other answers point out, there are numerous counterexamples to this strong claim. 
It is however likely that a weaker form of this statement is true. Singmaster's conjecture states that there exists some $M \lt \infty$ such that $f(t) \leqslant M$ for all $t$. This is still open. The best known upper bound, due to D. Kane, is $$f(t) = O \left(\frac{(\log t) \cdot (\log \log \log t) }{(\log \log t)^3} \right) .$$ On the other hand, it is consistent with our current knowledge that $f(t)$ never exceeds $8$. 
Reference: This MO post by Michael Hardy asks for recent progress on this problem. 
A: There are one or two (nontrivial) choices of $(m,n)$ where $\binom{n}{2} = \binom{m}{3}$.   Comparing the sequences on OEIS might be faster than coming up with the correct words to find it in a search engine. 
Edit: 120 is in both sequences, with $m=10$ and $n=16$.  In OEIS the list of triangular numbers ( http://oeis.org/A000217 ) and tetrahedral numbers ( http://oeis.org/A000292 ) shows that there are no other small examples and I am pretty sure that this pair is known to be the largest integer solution of $3y(y-1)=x(x-1)(x-2)$, and that this was first proved in an elementary way that does not use the theory of elliptic curves.
A: 10-choose-4 = 21-choose-2.${}{}{}{}$
A: $$\binom{10}{3}=\binom{16}{2}.$$
