Background checks are a consumer fraud. The Feds rarely prosecute those who - being “prohibited” still try to acquire a firearm from a Federally-licensed dealer. The attempt violates 18 U.S. Code § 922 (a)(6). My conclusion is based on public and published sources, to which I cite.
Only 62 Federal prosecutions followed 76,142 denials (in 2010) of purchase applicants. For the data see, Regional Justice Information Service, Enforcement of the Brady Act, 2010 , 2012, p. 7 https://www.ncjrs.gov/pdffiles1/bjs/grants/239272.pdf .
A Government Accountability Office (GAO) Report (No. 18-440, Sept., 2018) shows that of 112,090 denials by FBI screeners, only 12,710 were sent for “investigation”. Of that number only 12 resulted in prosecutions!! Plainly, far more than 12 denials were fully justified.
It is a Federal felony for a “prohibited person” to possess or to try to acquire any firearm. So few prosecutions show that Federal authorities do little about stopping those, who seek to abuse firearms.
As at end-2018, there were some 423,000,000 firearms in the U.S., so it seems reasonable to assume a denied person - determined to get a firearm - will do so. For the number of firearms see: U.S. Department of Justice, Firearms Commerce in the United States , 2000 and 2019; these data exclude military firearms. The U.S. at end-2018 had about 328,000,000 residents.
Were even half of those properly denied to be prosecuted by U.S. Attorneys, they would have little time to prosecute other Federal crimes, e.g., espionage, securities fraud, drug trafficking, etc. See: U.S. Department of Justice, U.S. Attorneys’ Annual Statistical Report - Fiscal Year 2019 , p. 4. See: Annual Statistical Reports (justice.gov) .
It is thus clear that we’d have to hire an army of prosecutors and build many new Federal prisons to hold those who are so determined to get a weapon, that they’ll risk a prison term.