In this paper we show analytically that the 2-norm polynomial numerical hulls of degrees 1 through n−1 for an n by n Jordan block are disks about the eigenvalue with radii approaching 1 as n→∞, and we prove a theorem characterizing these radii r_k,n. In the special case where k=n−1, this theorem leads to a known result in complex approximation theory: For n even, r_n−1,n is the positive root of 2rⁿ+r−1=0, and for n odd, it satisfies a similar formula. For large n, this means that r_n−1,n≈1−log(2n)/n+log(log(2n))/n. These results are used to obtain bounds on the polynomial numerical hulls of certain degrees for banded triangular Toeplitz matrices and for block diagonal matrices with triangular Toeplitz blocks.