This paper explores flow patterns in corporate bond mutual funds. We show that corporate bond funds exhibit a concave flow-to-performance relationship: their outflows are sensitive to bad performance more than their inflows are sensitive to good performance. Moreover, corporate bond funds tend to have greater sensitivity of outflows to bad performance when they have more illiquid assets and when the overall market illiquidity is high. These results point to the possibility of fragility in the fast-growing corporate bond market. The illiquidity of corporate bonds may generate a first-mover advantage among investors in corporate bond funds, amplifying their response to bad performance.