For the most part, the biggest holders of any country's sovereign debt are that country's own banking system. So, when the government comes under pressure in the sovereign debt markets, and the value of its debt decreases, it stresses the banks, who are holding so much of that debt on their own balance sheets.
In turn, when the banks have to be bailed out, the cost falls largely on the public. This increases the country's government debt-to-GDP ratio, causing its sovereign bonds to come under further pressure. It's a vicious cycle.
One of the big reasons banks can come under pressure in the first place, though, is because of deposit outflows.