Should you be able to insure Bitcoin the way banks insure bank accounts? If insurance for Bitcoin were available for Bitcoin losses what kind of losses would the insurance cover?
It is covered by a cyber-insurance policy, the kind taken out by businesses that need to provide proof of security audits, etc. in order to get the insurance. NAIC lists things that would be covered (http://www.naic.org/cipr_topics/topic_cyber_risk.htm), which includes;
"Theft of valuable digital assets, including customer lists, business trade secrets and other similar electronic business assets." These policies are not typically offered to individuals.

The problem, and the reason why this is true, is that these policies are very expensive, relative to cost, and there is a adverse selection problem in insurance (https://en.wikipedia.org/wiki/Adverse_selection), as well as moral hazard (https://en.wikipedia.org/wiki/Moral_hazard) - so insurers won't ever fully cover these small risks that they can't observe or protect against, because they will end up losing money on them.