Using angle-resolved photoemission, we report the first observation of an incomplete charge-density-wave (CDW) gap opening for the quasi-two-dimensional material Mo4O11. Our data indicate that the gap opening occurs over a range of 0.5 (A)ngstrom(-1) in the Brillouin zone. The maximum gap opening (Delta(max)) we observe is 60 +/- 10 meV, which leads to a ratio of 2 Delta(max)/kT(CDW) = 14 +/- 6 (T-CDW = transition temperature), well above the weak coupling value of 3.5. Our data suggest that both 1-dimensional and 2-dimensional electronic states may be involved in the gap opening.