[Haskell], 102 98 bytes

import Data.List
f(x:r)|length r<x||x<0=0>1|1<3=f.reverse.sort$(pred<$>take x r)++drop x r
f x=1<3

Try it online! Usage: f [3,3,2,2,1,1], returns True or False. Assumes that the input contains no zeros and is sorted in descending order, as allowed in the challenge.

Explanation:

import Data.List                 -- import needed for sort
f (x:r)                          -- x is the first list element, r the rest list
  |length r < x || x<0 = False   -- if the rest list is shorter than x or x<0, return False
  |otherwise    = f .            -- else call f recursively
      reverse . sort $           -- with the descendingly sorted list ...
      (pred<$>take x r)          -- of the first x elements of r subtracted by 1 and ...
      ++ drop x r                -- the rest of r
f [] = True                      -- if the list is empty, return True

Edit: This seems to follow the Havel-Hakimi mentioned in other answers, though I did not know of this algorithm when writing the answer. [Haskell]: https://www.haskell.org/