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.

Note that, at least the arXiv reference above, deals with subschemes of $\mathbb{A}^d$ not necessarily supported at the origin.  On the other hand, since for $n \leq 8$ the above is the only non-irreducible example, the "new" component must be supported at the origin.  Indeed in the case *n=8* and *d=4*, the extra component is a product of a Grassmannian and an affine space.