No. Any reputable insurance company would not backdate your policy. Some consumers do request that insurance agents post-date or back date policies sometimes so that they can show proof of insurance to a court when they actually had a lapse. The fact is that no auto insurance company should backdate a car insurance policy, period. In general backdating the effective date of an auto insurance policy is not permitted.
If an insurance carrier would backdate the start date of a policy this would inevitably favor the insured, because it opens the insurance company to risk. For example a person could have been in an accident during this time and though they did not have insurance now try to claim it if the insurance was post-dated.
Now if you had been without insurance for a few days you may have been able to reinstate your policy. Most auto insurance carriers will reinstate a policy cancelled for reasons such as non-payment, as long as the lapse in coverage is less than 30 days, and there are no losses or claims. When such as reinstatement is done you must normally sign a statement of no loss so that you verify there were no incidents during the lapse and you could and would not make a claim for that period of time.
If you were cited by police or found by your Department of Motor Vehicles to be without auto insurance an insurance company cannot backdate a policy to show proof of insurance since you had none at that point. You can see if buying insurance before your court date would help you at all in court though by showing you are now an insured motorist acting responsibly.
Penalties for driving without insurance typically include a large fine and possible jail time and in some states also include suspension of your license plates (registration) and driver's license. It is important to not drive without insurance or to have a lapse in insurance either since many states will also penalize you for a gap in coverage with penalties such as the suspension or revocation of your registration for a certain period of time.