An n-by-n real symmetric matrix is said to be a Hankel matrix if , for each and . The symmetric nonnegative inverse eigenvalue problem (SNIEP) asks when a list of real numbers is the spectrum of an n-by-n symmetric nonnegative matrix H. In this paper, we search for conditions on the list for the matrix H to be Hankel. For n = 3, sufficient conditions are established. In particular, a necessary and sufficient condition is obtained if Λ is a list of three nonnegative numbers. Also, if , we give conditions for realizability by a Hankel matrix. Finally, we present a special type of list that can serve as the spectrum of a Hankel nonnegative matrix with positive trace. Several of our results are constructive and provide a Hankel realizing matrix.