We systematically analyse how network structure and bank characteristics affect solvency distress contagion risk in interbank networks. As interbank networks become more connected and more regular in structure, the contagion risk of system-wide shocks and individual bank defaults initially increases and then decreases, all else being equal. The low density heterogeneous network structures that typify real interbank networks are particularly prone to solvency distress contagion risk, when banks are similar in balance sheet size and capitalisation. However, when networks are calibrated to UK data, the higher capitalisation of large, highly-connected banks relative to their interbank exposures significantly increases the resilience of the system and reduces the importance of network structure. These findings reinforce the importance and effectiveness of imposing higher capital buffers on systemically important banks and of policies that limit interbank exposures. We also demonstrate that for real-world interbank networks, simple network metrics other than individual bank connectedness do not provide robust indicators for monitoring solvency contagion risk, suggesting that policymakers should continue efforts to model these risks explicitly rather than rely on simple aggregate indicators.