Complex mutual dependencies of asset returns are recognized to contribute to systemic risk. A growing literature emphasizes that identification of vulnerable firms is a fundamental requirement for mitigating systemic risk in a given asset market. However, in reality, firms are generally active in multiple asset markets with potentially different degrees of vulnerabilities in different markets. Therefore, to assess combined risks of the firms, we need to know how systemic risk measures of firms are related across markets? In this paper, we answer this question by studying US firms that are active in both stock as well as corporate bond markets. The main results are twofold. One, firms that exhibit higher systemic risk in the stock market also tend to exhibit higher systemic risk in the bond market. Two, systemic risk within an asset category is related to firm size, indicating that `too-big-to-fail' firms also tend to be `too-central-to fail. Our results are robust with respect to choose of asset classes, maturity horizons, model selection, time length of the data as well as controlling for all major market level factors. These results have prominent policy implications for identification of vulnerabilities and targeted interventions in financial networks.