You can, but they stated the insurance companies would argue more the more claims you file. Ram a ship and they replace your ship free in 12 hours. Destroy the next ship and it takes 24, then 72, a week, etc.
You can also lose it if you are caught using it for fraud. Take your ship out and let a buddy pirate capture it, report the loss, reclaim ship and do it again = fraud
Lots of stuff still up the air and unexplained.
As for the ships, I pledged some absurd amount, $175 or something, for the massive cargo ship that was available for a limited time. The thing could carry a ton on goods, 50x what a starter ship could or something, potentially earning much more, much faster. I'm probably more inclined to risk more dangerous area's of space if all I have to worry about losing is my cargo. Who knows at this point, it may mean nothing, but meh. Even if some bigger ships come out that trump this one, I can always just hire NPCS to fly this one around in my fleet, or with other NPC escorts.
Lastly, the way the ship insurance works is that it covers the chassis of your ship, the default. Buy a new engine and tons of guns? All gone. I believe you can insure that stuff separately, but yeah.
Also, typed all of this on my phone, so sorry for a lack of elaboration/abundunce of errors!