IfA is a regular local ring of dimensionr>2, over an algebraically closed fieldk, we show that the Hilbert scheme Hilb n A parametrizing ideals of colengthn inA(dim k A/I=n) has dimension>cn 2−2/r and is reducible, for alln>c′, wherec andc′ depend only onr. We conclude that ifV is a nonsingular projective variety of dimensionr>2, the Hilbert scheme Hilb n V parametrizing the 0-dimensional subschemes ofV having lengthn, is reducible for alln>c″(r). We may takec″(r) to be (1) $$102 ifr = 3,25 ifr = 4,35 ifr = 5,and\left( {1 + r} \right)\left( {{{1 + r} \mathord{\left/ {\vphantom {{1 + r} 4}} \right. \kern-\nulldelimiterspace} 4}} \right)ifr > 5.$$ The result answers in the negative a conjecture of Fogarty [1] but leaves open the question of the conjectured irreducibility of Hilb n A, whereA has dimension 2. Hilb n V is known to be irreducible ifV is a nonsingular surface (Hartshorne forP 2, and Fogarty [1]). In all cases Hilb n V and Hilb n A are known to be connected (Hartshorne forP r, and Fogarty [1]). The author is indebted to Hartshorne for suggesting that Hilb n A might be reducible ifr>2. The proof has 3 steps. We first show that ifV is a variety of dimensionr, then Hilb n V is irreducible only if it has dimensionr n. We then show that ifA is a regular local ring of dimensionr, Hilb n A can be irreducible only if it has dimension (r−1)(n−1). Finally in § 3 we construct a family of graded ideals of colengthn in the local ringA, and having dimensionc′ n2−2/r. Since for largen this dimension is greater thanr n, and since Hilb n A↪Hilb n V whenA is the local ring of a closed point onV, the proof is complete, except for (1), which follows from § 3, and the monotonicity of (dim Hilb n V−r n) (see (2)). In § 4, we comment on some related questions.