State laws differ but in general you cannot possess two different state's driver's license at the same time.
Normally you need to be licensed in the state that you are considered to have permanent residency in and that license would be good to use throughout the United States to drive as long as it is valid. However there are some exceptions to this general rule and thus you are allowed to obtain a second license without surrendering your primary license.
Usually two driver's license are not allowed since it can be confusing to keep an accurate driving record for the person or a state does not want to license you so that you can avoid suspensions, penalties, etc in the other state. That is why no matter what type of license you are applying for the state's licensing office will check with the National Driver Registry (NDR). This lists problem drivers who have serious offenses on their record such as a DUI or whose license is suspended, revoked, canceled, etc.
The one state that we are aware of that allows you to obtain a second license is Florida. The Florida Department of Highway Safety and Motor Vehicles (HSMV) states that part-time residents may elect to retain their valid out-of-state license and receive a Florida license which states "Valid in Florida Only." So in most all states one has to turn in or exchange their previous state's license for a new license when they move to a new state but FL allows you to keep your other state's license and carry a Florida only license.
What allows this is Florida Statute (FS) Section 322.03(1)(B) which states that a person who does not drive a commercial motor vehicle is not required to surrender a license issued by another jurisdiction, upon a showing to the department that such license is necessary because of employment or part-time residence.
This statute goes on to say that any person who retains a driver's license because of employment or part-time residence shall, upon qualifying for a license in this state, be issued a driver's license which shall be valid within this state only. A person may not have more than one valid Florida driver's license at any time.
If a person does turn in their license to get a Florida driver's license then FL law notes that all surrendered licenses may be returned by the department to the issuing jurisdiction together with information that the licensee is now licensed in a new jurisdiction or may be destroyed by the department, which shall notify the issuing jurisdiction of such destruction.
While most states require you to surrender a driver's license if you obtain a new license there Florida allows you to retain your "home state" license when living part-time in Florida and have a need to keep your other driver's license but obtain a FL only license. Other states may have similar laws but you would need to check with the Department of Motor Vehicles of specific state in which you want to find out if two licenses are allowed since most states do not allow this.
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Comment update: When this question was originally asked, there was a "Florida only" license available. However, effective November 1, 2009, a Florida law was passed that prohibits the HSMV from issuing a "Valid in Florida Only" license. This change along with a few other changes regarding Florida driver's licenses were made in order to comply with the Real ID Act mandates. [Let CarInsurance.com help you find affordable auto insurance now.] | 
