Multivalent adhesive interactions mediated by a large number of ligands and receptors underpin many biological processes, including cell adhesion and the uptake of particles, viruses, parasites, and nanomedical vectors. In materials science, multivalent interactions between colloidal particles have enabled unprecedented control over the phase behavior of self-assembled materials. Theoretical and experimental studies have pinpointed the relationship between equilibrium states and microscopic system parameters such as the ligand-receptor binding strength and their density. In regimes of strong interactions, however, kinetic factors are expected to slow down equilibration and lead to the emergence of long-lived out-of-equilibrium states that may significantly influence the outcome of self-assembly experiments and the adhesion of particles to biological membranes. Here we experimentally investigate the kinetics of adhesion of nanoparticles to biomimetic lipid membranes. Multivalent interactions are reproduced by strongly interacting DNA constructs, playing the role of both ligands and receptors. The rate of nanoparticle adhesion is investigated as a function of the surface density of membrane-anchored receptors and the bulk concentration of nanoparticles and is observed to decrease substantially in regimes where the number of available receptors is limited compared to the overall number of ligands. We attribute such peculiar behavior to the rapid sequestration of available receptors after initial nanoparticle adsorption. The experimental trends and the proposed interpretation are supported by numerical simulations.