We report results from new experiments on C/Re(11 (2) over bar1) to identify threshold conditions for morphological instability of Re(11 (2) over bar1). We have found that adsorption of carbon from 0.35 to 0.85 ML (0.3-6.0 L exposure of C2H2) at T >= 800 K leads to faceting of Re(11 (2) over bar1) with formation of three-sided pyramids. Using density functional theory we have investigated binding sites and binding energies of C on planar and faceted Re surfaces as well as generated a surface phase diagram of C/Re to obtain an atomistic understanding of C-induced pyramidal faceting of Re(11 (2) over bar1). The calculations reveal that at low to intermediate coverage, C atoms prefer binding at four-fold sites on the Re surfaces and formation of three-sided pyramids is thermodynamically favored.