Anti-Gunrunning Act of 2001

Anti-Gunrunning Act of 2001 - Amends the Brady Handgun Violence Prevention Act to prohibit any licensed firearms importer, manufacturer, or dealer from selling, delivering, or transferring: (1) two or more handguns to any single person (other than a licensed importer, manufacturer, or dealer) during any 30-day period; or (2) a handgun knowing or having reasonable cause to believe that the transferee has already received one or more handguns within the previous 30 days. Prohibits an unlicensed individual from receiving more than one handgun within any 30-day period. Specifies exceptions.

Provides for imprisonment for up to five years (currently, one year) of a licensed dealer, importer, manufacturer, or collector knowingly making any false statement in connection with required firearms records.

Extends the deadline for the destruction of records relating to handgun transfers subject to the waiting period from 20 business days to 35 calendar days after the date the transferee made the statement on the basis of which notice of the transaction was provided to the chief law enforcement officer of the transferee's place of residence. Requires the national instant criminal background check system, if receipt of a firearm would not violate Federal or State law, to destroy records relating to the person or the transfer within 35 calendar days after the system provides the licensee with the identification number unique to the transfer.