We investigate whether surfaces that are complete intersections of quadrics and complete intersection surfaces in the Segre embedded product \mathbf{P}^1\times \mathbf{P}^k\hookrightarrow \mathbf{P}^{2k+1} can belong to the same Hilbert scheme. For k=2 there is a classical example; it comes from K3 surfaces in projective 5 -space that degenerate into a hypersurface on the Segre threefold. We show that for k\geq 3 there is only one more example. It turns out that its (connected) Hilbert scheme has at least two irreducible components. We investigate the corresponding local moduli problem.