Nanocomposite films containing a high volume fraction (>50 vol %) of nanoparticles (NPs) in a polymer matrix are promising for their functionality and use as structural coatings and also provide a unique platform to understand polymer behavior under strong confinement. Previously, we developed a novel technique to fabricate such nanocomposites at room temperature using solvent-driven infiltration of polymer (SIP) into NP packings. In the SIP process, a bilayer made of an underlying polymer film and a dense packing of NPs is exposed to solvent vapor which induces condensation of the solvent into the voids of the packing. The condensed solvent plasticizes the underlying polymer film, inducing polymer infiltration into the solvent-filled voids in the NP packing. In this work, we study the effect of confinement on the kinetics of SIP and the final partitioning of the polymer into the interstices of the NP packing. We find that, while the dynamics of infiltration during SIP are strongly dependent on confinement, the final extent of infiltration is independent of confinement. The time for infiltration obeys a power law with confinement, as defined by the ratio of the chain size and the pore size. Qualitatively, the observed time scale is attributed to changes in concentration regimes as infiltration proceeds, which lead to shifting characteristic length scales in the system over time. When the concentration in the pore exceeds the critical overlap concentration, the characteristic length scale of the polymer is no longer that of the entire chain but rather the correlation length, which is smaller than the pore size. Therefore, at long times, the extent of infiltration is independent of the confinement ratio. Furthermore, favorable surface interactions between the polymer and the NPs enhance partitioning into the NP packing.