The adhesion of a living cell to an extracellular matrix surface is effected through the bonding of receptor molecules in the cell membrane to compatible ligand molecules on the surface. In a series of experiments on adhesion of cells to a substrate surface with a controlled density of ligand binding sites, Arnold et al. (ChemPhysChem 5:383, 2004) showed that tight cell adhesions could form only if the areal density of binding sites on the substrate was higher than some critical value. Furthermore, this critical value was consistent across the four cell types examined in the experiments. For ligand density below the critical level, on the other hand, virtually no adhesions formed. In this article, we examine the competition between thermal undulations of the cell membrane and its adhesion to the substrate. In particular, we show that thermal undulations destabilize membrane bonding to the substrate unless the bond spacing is below a certain level. By following this line of reasoning in the context of classical statistical mechanics, we obtain an estimate of the critical value of spacing which is in reasonable agreement with the observations.