The TRAPPIST-1 system provides an extraordinary opportunity to study multiple terrestrial extrasolar planets and their atmospheres. Here, we use the National Center for Atmospheric Research Community Atmosphere Model version 4 to study the possible climate and habitability of the planets in the TRAPPIST-1 system. We assume the worlds are ocean-covered, with atmospheres composed of N2, CO2, and H2O, and with orbital and geophysical properties defined from observation. Model results indicate that the inner three planets (b, c, and d) presently reside interior to the inner edge of the traditional liquid water habitable zone. Thus, if water ever existed on the inner planets, they would have undergone a runaway greenhouse and lost their water to space, leaving them dry today. Conversely, the outer three planets (f, g, and h) fall beyond the maximum CO2 greenhouse outer edge of the habitable zone. Model results indicate that the outer planets cannot be warmed, despite having as much as 30 bar CO2 atmospheres, instead entering a snowball state. The middle planet (e) represents the best chance for a presently habitable ocean-covered world in the TRAPPIST-1 system. Planet e can maintain at least some habitable surface area with 0-2 bar CO2, depending on the background N2 content. Near-present-day Earth surface temperatures can be maintained for an ocean-covered planet e with either 1 bar N2 + 0.4 bar CO2, or a 1.3 bar pure CO2 atmosphere.