Once you are put on Brady bill list is the restriction life time?
The Official name is "The Brady Handgun Violence Prevention Act". The act mandated federal background checks and a five-day waiting period. When the NICS system was implemented in 1998 the 5 day waiting period was changed.
The act came about as a results of the attempted assassination of the President of the United States, Ronald Reagan in 1981 by John Hinkley, Jr.
During the assassination attempt of the President he shot James Brady.
Some of reasons you may be on the Brady list and prohibited from receiving a firearm and other provisions are below. In regard to how long you may be on the list please review Title 28, Chapter 1, Part 25 of the Code of Federal Regulations, which sets forth many regulations and provisions dealing the act.
Section 25-9 provides the following appears to say the records will be retained indefinitely unless they are canceled.
§ 25.9 Retention and destruction of records in the system.
(a) The NICS will retain NICS Index records that indicate that receipt of a firearm by the individuals to whom the records pertain would violate Federal or state law. The NICS will retain such records indefinitely, unless they are canceled by the originating agency. In cases where a firearms disability is not permanent, e.g., a disqualifying restraining order, the NICS will automatically purge the pertinent record when it is no longer disqualifying. Unless otherwise removed, records contained in the NCIC and III files that are accessed during a background check will remain in those files in accordance with established policy.
You may be placed on the list under the following under 18 U.S. Code Section 922 (d)
(d) It shall be unlawful for any person to sell or otherwise dispose of any firearm or ammunition to any person knowing or having reasonable cause to believe that such person—
(1) is under indictment for, or has been convicted in any court of, a crime punishable by imprisonment for a term exceeding one year;
(2) is a fugitive from justice;
(3) is an unlawful user of or addicted to any controlled substance (as defined in section 102 of the Controlled Substances Act (21 U.S.C. 802));
(4) has been adjudicated as a mental defective or has been committed to any mental institution;
(5) who, being an alien—
(A) is illegally or unlawfully in the United States; or
(B) except as provided in subsection (y)(2), has been admitted to the United States under a nonimmigrant visa (as that term is defined in section 101(a)(26) of the Immigration and Nationality Act (8 U.S.C. 1101(a)(26)));
(6) who  has been discharged from the Armed Forces under dishonorable conditions;
(7) who, having been a citizen of the United States, has renounced his citizenship;
(8) is subject to a court order that restrains such person from harassing, stalking, or threatening an intimate partner of such person or child of such intimate partner or person, or engaging in other conduct that would place an intimate partner in reasonable fear of bodily injury to the partner or child, except that this paragraph shall only apply to a court order that—
(A) was issued after a hearing of which such person received actual notice, and at which such person had the opportunity to participate; and
(i) includes a finding that such person represents a credible threat to the physical safety of such intimate partner or child; or
(ii) by its terms explicitly prohibits the use, attempted use, or threatened use of physical force against such intimate partner or child that would reasonably be expected to cause bodily injury; or
(9) has been convicted in any court of a misdemeanor crime of domestic violence.
This subsection shall not apply with respect to the sale or disposition of a firearm or ammunition to a licensed importer, licensed manufacturer, licensed dealer, or licensed collector who pursuant to subsection (b) of section 925 of this chapter is not precluded from dealing in firearms or ammunition, or to a person who has been granted relief from disabilities pursuant to subsection (c) of section 925 of this chapter.
If you are denied a gun wrongfully you have recourse.
18 U.S. Code Section 925A
Any person denied a firearm pursuant to subsection (s) or (t) of section 922—
(1) due to the provision of erroneous information relating to the person by any State or political subdivision thereof, or by the national instant criminal background check system established under section 103 of the Brady Handgun Violence Prevention Act; or
(2) who was not prohibited from receipt of a firearm pursuant to subsection (g) or (n) of section 922,
may bring an action against the State or political subdivision responsible for providing the erroneous information, or responsible for denying the transfer, or against the United States, as the case may be, for an order directing that the erroneous information be corrected or that the transfer be approved, as the case may be. In any action under this section, the court, in its discretion, may allow the prevailing party a reasonable attorney’s fee as part of the costs.
(Added Pub. L. 103–159, title I, § 104(a), Nov. 30, 1993, 107 Stat. 1543.)
United Stated Code https://www.law.cornell.edu/uscode/text/18/part-I/chapter-44
Code of Federal Regulations dealing with registration of people on the Brady List and other provisions. https://www.law.cornell.edu/cfr/text/28/part-25/subpart-A
Wikipedia Article. https://en.wikipedia.org/wiki/Brady_Handgun_Violence_Prevention_Act