One striking example of the failure of Hamilton-Ivey pinching can be seen <a href="http://math.stanford.edu/~maximo/Maximo%20-%20On%20the%20blow-up%20of%20four%20dimensional%20Ricci%20flow%20singularities.pdf">here</a> in which it is shown that the <a href="http://projecteuclid.org/euclid.jdg/1090511686">FIK shrinkers</a> (which do not have non-negative Ricci curvature, much less sectional curvature, can arise as blowup limits to the Ricci flow. 

As far as I know, basically all that is known is that ancient solutions to the Ricci flow (e.g. blowup limits) have non-negative scalar curvature, but <a href="http://arxiv.org/pdf/0706.3081.pdf">Corollary 2.5</a>