Yes, auto insurance companies can deny coverage to drivers. Insurance companies do not like high-risk drivers, so if they think covering you means they are highly likely to have to pay a claim on your behalf, they can deny you coverage. For example, they can deny you insurance for having multiple tickets, accidents, serious violations like DUI, or even a low credit score.
If you are denied coverage by a standard company such as Allstate or Progressive, you can shop with non-standard or high-risk insurers to find a policy. Your policy will definitely cost more, but at least you will be insured and able to drive. The main thing is to try to correct your driving habits over time so that you can get insurance at a better rate in the future. Time heals all!