The U.S. Department of Justice just released how accurate the background check is for firearm dealers to use when people try to buy firearms. 

In their report (can be read here), the NICS system is almost 100%

We found the FBI generally has an effective internal control system and quality control process. The FBI processed more than 51 million NICS transactions from 2008 to 2014, and approved or denied about 50 million of them (state authorities processed more than 68 million requests during the same period). The FBI denied 556,496 of these transactions. The FBI told us its quality control process results in an accuracy rate that ranges from 99.3 percent to 99.8 percent.

We judgmentally selected 447 denied transactions and found that only 1 transaction was incorrectly denied, resulting in a 99.8 percent accuracy rate.