This paper studies the elementary symmetric polynomials S_k(x) for x \in \mathbb{R}^n. We show that if |S_k(x)|,|S_{k+1}(x)| are small for some k>0 then |S_\ell(x)| is also small for all \ell > k. We use this to prove probability tail bounds for the symmetric polynomials when the inputs are only t-wise independent, that may be useful in the context of derandomization. We also provide examples of t-wise independent distributions for which our bounds are essentially tight.