It's a good point, so I did a quick google search and it appears that in most countries the insurance is paid out by a seperate insurance fund that all banks pay in to over time. Presumably the fund is reinsured with lloyds, Berkshire Hathaway or something similar. So it's independent of the banks and the state.
I couldn't find anything specifically for Cyprus, so I'm not sure what the situation is there, but it's probably not much different from other countries.
Edit: Here is a good explanation: http://europa.eu/rapid/press-release_MEMO-10-318_en.htm?loca...
"7) Will Deposit Guarantee Schemes have enough funds to pay out in case banks fail?
There have been shortcomings in some countries in the past. It is not feasible or necessary to provide schemes with an amount of money equivalent to all deposits. But banks will have to pay on a regular basis to the schemes, in advance, so a pot of money can be built up, and not only after a bank failure. Such 'ex-ante funds' will make up 75% of the overall funds in DGS.
If it becomes necessary, banks will have to pay additional contributions, which will contribute a further 25% of the target funds. If this is still insufficient, Deposit Guarantee Schemes could borrow from each other ("mutual borrowing facility") up to a certain limit (again 25% of target funds) or use additional funding sources such as borrowing on the financial market, e.g. by issuing bonds.
The new financing requirements will ensure that each scheme has enough funds in place to deal with a medium-size bank failure. This level is comparable to the existing well-financed schemes in the EU. These levels of funding will have to be achieved in all Member States by 2020.
Banks having a riskier business model than others will pay higher contributions to Deposit Guarantee Schemes - up to about 3 times more."