Robert Baillie has a paper on arxiv today (http://arxiv.org/abs/1105.3943) which shows how in principle one can construct examples of formulae which hold for $N=0,1,2,\ldots,k$, for arbirtrarily large $k$, then fail for all larger $N$.
His largest example holds with $k\approx \exp(\exp(\exp(\exp(\exp(\exp(e))))))$.
