There are a few states that now have made it possible for law enforcement to have access to the DMV and insurance records to make sure you are currently insured however you are normally still required to carry proof of insurance (your insurance card) with you in your vehicle.
For example the California Department of Motor Vehicles (DMV) notes that as of January 1, 2006, all insurance companies are required to report insurance status information to DMV for all private use vehicles (CA Vehicle Code 16058).
As of July 1, 2006, law enforcement and court personnel have access to DMV records to verify that your California registered vehicle is currently insured (CVC 16058.1).
Effective October 1, 2006, your vehicle registration is subject to suspension if the liability insurance is cancelled, OR if your insurance company has not electronically provided evidence of insurance when you purchase and register your vehicle, OR if you provide DMV with false insurance information (CVC 4000.38).
The CA DMV though specifically and emphatically states that even with these laws in place one must still carry proof of insurance in their vehicle at all times.
Texas also has a new insurance verification system in place and the TX Department of Public Safety (DPS) notes that yes, an insurance company must still issue an insurance card to insureds and that drivers must still carry the card in the vehicle.
Texas notes that while the objective of this new verification program is to provide accurate and timely proof of insurance via an electronic database system, it is still required by law (601.053 of the Texas Transportation Code) to make available current effective paper proof of insurance upon request.
To find out about your specific state's requirements regarding proof of insurance one must carry with them, and if it is necessary with a statewide verification program available to authorities, check with your local Department of Motor Vehicles. Likely you will still need your insurance card on you and shown to authorities upon request, especially if say the insurance database is down or has technical difficulties so that the police were unable to use it during their traffic stop.