The NBA playoffs have been a part of the season since 1946, and no team has ever been able to overcome a three game deficit in a series.
Specifically, teams have a 0-150 overall record when down 0-3 in a playoff series, per SportsCenter.
The closest an NBA team has come;
2003 NBA Playoffs
First Round
(3) Dallas Mavericks vs. (6) Portland Trail Blazers
Dallas lead the series 3 - 0, Portland would go on to tie the series 3 - 3, with a Game 7 in Dallas.
These were the main participants for Game 7
Portland Trail Blazers
Damon Stoudamire
Bonzi Wells
Rasheed Wallace
Zach Randolph
Arvydas Sabonis
Rubin Patterson
Scottie Pippen
———
Dallas Mavericks
Steve Nash
Michael Finley
Eduardo Najera
Dirk Nowitzki
Raef LaFrentz
Nick Van Exel
Shawn Bradley
———
At the end of 3 quarters, Portland were up 73 - 71. But Dallas would score 36 in the 4th to secure the game and the series.