Bankruptcy of an insurance company means that their debts exceed their assets. That means that their shareholders lose everything (i.e., get nothing back).
There are basically two ways this can happen: (i) losses (claims) exceed premiums (and that has to be often enough and by a sufficiently large margin to exceed the insurer's capital and surplus) or (ii) their investment portfolio goes bust - that happened to a number of companies that were heavily invested in asset backed securities when the market tanked.
Of course there are insurance companies that guarantee the performance of investments - i.e., issue guarantee bonds of one sort or another and they got nailed when the market tanked - and others that guarantee that companies will complete projects on time (Surety bonds).
Speaking of the market tanking - it only took a Gaussian copula formula to do it
In the mid-’80s, Wall Street turned to the quants—brainy financial engineers—to invent new ways to boost profits. Their methods for minting money worked brilliantly… until one of them devastated the global economy.
www.wired.com
BTW, the formula wasn't wrong - the problem is the people using it didn't really understand what it meant or how it worked.