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In this paper, we show that a set of q+a hyperplanes, q>13, a<(q-10)/4, thatdoes not cover PG(n,q), does not cover at least q^(n-1)-aq^(n-2) points, andshow that this lower bound is sharp. If the number of non- covered points is atmost q^(n-1), then we show that all non-covered points are contained in onehyperplane. Finally, using a recent result of Blokhuis, Brouwer, and Szonyi[3], we remark that the bound on a for which these results are valid can beimproved to a<(q-2)/3 and that this upper bound on a is sharp

Partial covers of PG(n,q)