It is certainly not irreducible if n=8 and d>3. This is analyzed nicely in the paper
Hilbert schemes of 8 points, Dustin A. Cartwright, Daniel Erman, Mauricio Velasco, Bianca Viray available at http://arxiv.org/abs/0803.0341 (and I think published in ANT).
From there you can look at the references, especially I think the first paper studying in detail this problem was
Anthony Iarrobino. Reducibility of the family of 0-dimensional schemes on a variety. Inventiones Math., 15:72–77, 1972.