If you have previously been banned or currently have lots of penalty points you can still get insurance but most likely at a higher rate.
No claims bonuses accrued through driving a company car normally won't be valid if you change to private insurance. Unless the policy os your own and not aquired through your employer. If you currenly drive a company car and are thinking about leaving the company, then why not get some quotes today?
When making a claim have as many details with you as possible including your policy number and a crime reference number if you have one.
If you own a "classic car" you may be able to get cheap insurance if you only do limited mileage each year. Often many companies have special rates for thsi type of vehicle especially if it is not your main car.