Is there a number k such that every natural number can be written as $\sum_{i=1}^k \binom{a_i}{3}$ for some natural numbers $a_i$'s?
Watson's nice paper "Sums of eight values of a cubic polynomial" (http://jlms.oxfordjournals.org/content/s127/2/217.full.pdf) shows that we may take $k = 8$.
It is an old conjecture of Pollock that every positive integer is the sum of at most five tetrahedral numbers. In fact it is conjectured that all positive integers except for a finite list of 241 numbers, A000797, are the sum of $4$ tetrahedral numbers.

$\begingroup$ Gjergji: Is it known that there exists a finite number $k$ such that each number is a sum of $k$ tetrahedral numbers? $\endgroup$ – TOM Feb 11 '15 at 1:11

$\begingroup$ Yes, that's known for any polynomial that takes integer values. $\endgroup$ – Gjergji Zaimi Feb 11 '15 at 2:07

1$\begingroup$ The earliest reference I found is: Kamke, E.. "Verallgemeinerungen des WaringHilbertschen Satzes." Mathematische Annalen 83 (1921): 85112 $\endgroup$ – Gjergji Zaimi Feb 11 '15 at 2:39

$\begingroup$ As Gjergji has pointed out, this is a consequence of a general theorem of Kamke. A modern reference is Nathanson's Elementary methods in Number Theory; see Chapter 11, especially Theorems 11.10 and 11.12. $\endgroup$ – socalled friend Don Feb 11 '15 at 3:04
In more general case this problem was studied by Nechaev, see