At the core of Saccharomyces cerevisiae telomeres is an array of tandem telomeric DNA repeats bound site-specifically by multiple Rap1 molecules. There, Rap1 orchestrates the binding of additional telomere-associated proteins and negatively regulates both telomere fusion and length homeostasis. Using electron microscopy, viscosity, and light scattering measurements, we show that purified Rap1 is a monomer in solution that adopts a ringlike or C shape with a central cavity. Rap1 could orchestrate telomere function by binding multiple telomere array sites through either cooperative or independent mechanisms. To determine the mechanism, we analyze the distribution of Rap1 monomers on defined telomeric DNA arrays. This analysis clearly indicates that Rap1 binds independently to each non-overlapping site in an array, regardless of the spacing between sites, the total number of sites, the affinity of the sites for Rap1, and over a large concentration range. Previous experiments have not clearly separated the effects of affinity from repeat spacing on telomere function. We clarify these results by testing in vivo the function of defined telomere arrays containing the same Rap1 binding site separated by spacings that were previously defined as low or high activity. We find that Rap1 binding affinity in vitro correlates with the ability of telomeric repeat arrays to regulate telomere length in vivo. We suggest that Rap1 binding to multiple sites in a telomere array does not, by itself, promote formation of a more energetically stabile complex.