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.