  We find a Penrose limit of AdS_5 x T^{1,1} which gives the pp-wave geometry
identical to the one that appears in the Penrose limit of AdS_5 x S^5. This
leads us to conjecture that there is a subsector of the corresponding N=1 gauge
theory which has enhanced N=4 supersymmetry. We identify operators in the N=1
gauge theory with stringy excitations in the pp-wave geometry and discuss how
the gauge theory operators fall into N=4 supersymmetry multiplets. We find
similar enhancement of symmetry in some other models, but there are also
examples in which there is no supersymmetry enhancement in the Penrose limit.
