I'll answer this along with the OP that mentions concentration:
1) Concentration does affect the emission lifetimes (and the quantum yield).
2) Size of the crystal does affect the emission lifetimes depending on the concentration.
1) Is called "concentration quenching". For example, you mentioned Nd, which has multiple 4f-state levels. Energy transfer between multiple Nd(III) ions could doubly excite a Nd(III) ion, leading to an emission of higher energy (and not 1064 nm line used for lasing). This usually lead to shorter emission lifetimes. There could also be a "killer-site", which could be a impurity or defect site that completely quenches the energy once the energy is delivered to this site, but most laser medium is very pure.
2) Is called "self-trapping". This is basically a reabsorption of a emitted photon by a different ion of the same type. Obviously, the larger the crystal, there are more dopant site that can absorb the photon. This causes the emission lifetimes to be come longer, and also decreases quantum yield.
This paper is relevant for this discussion:
F. Auzel, G. Baldacchini, L. Laversenne, G. Boulon, "
Radiation trapping and self-quenching analysis in Yb3+, Er3+, and Ho3+ doped Y2O3", Opt. Mater. 2003, 24, 103-109.Most of the laser medium are optimized in terms of dopant concentration for a particular size of crystal so that they can afford the maximum quantum efficiency possible for a given laser setup.
EDIT: Both the effects explained above is probably a small deviation in a high-purity solids like laser medium.