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Proteins and nucleic acids can spontaneously self-assemble into membraneless droplet-like compartments, both in vitro and in vivo. A key component of these droplets are multi-valent proteins that possess several adhesive domains with specific interaction partners (whose number determines total valency of the protein) separated by disordered regions. Here, using multi-scale simulations we show that such proteins self-organize into micro-phase separated droplets of various sizes as opposed to the Flory-like macro-phase separated equilibrium state of homopolymers or equilibrium physical gels. We show that the micro-phase separated state is a dynamic outcome of the interplay between two competing processes: a diffusion-limited encounter between proteins, and the dynamics within small clusters that results in exhaustion of available valencies whereby all specifically interacting domains find their interacting partners within smaller clusters, leading to arrested phase separation. We first model these multi-valent chains as bead-spring polymers with multiple adhesive domains separated by semi-flexible linkers and use Langevin Dynamics (LD) to assess how key timescales depend on the molecular properties of associating polymers. Using the time-scales from LD simulations, we develop a coarse-grained kinetic model to study this phenomenon at longer times. Consistent with LD simulations, the macro-phase separated state was only observed at high concentrations and large interaction valencies. Further, in the regime where cluster sizes approach macro-phase separation, the condensed phase becomes dynamically solid-like, suggesting that it might no longer be biologically functional. Therefore, the micro-phase separated state could be a hallmark of functional droplets formed by proteins with the sticker-spacer architecture.