Here's why White House Down 2 didn't happen. The original Die Hard was based on the 1979 novel Nothing Lasts Forever, about a retired cop who becomes trapped in a skyscraper during a terrorist takeover. The book was a sequel to author Roderick Thorp's The Detective, which was adapted into a Frank Sinatra movie of the same name. A contractual quirk meant Sinatra had to be offered Die Hard first, though he had retired from acting by this point. Everyone from Arnold Schwarzenegger to Richard Gere was offered the movie before Bruce Willis was hired for the lead role of John McClane.
Die Hard was a hit during its original release and is now regarded as an action classic. It also provided a template for action movies for the next decade, with many adopting the "Die Hard on an X" formula, such as Speed, Passenger 57 and Under Siege. Willis also returned for four sequels as McClane in the years that followed, with the most recent being 2013's critically reviled A Good Day To Die Hard.
There's a unique phenomenon that happens every few years in Hollywood where movies with very similiar premises come along at the same time. This includes 1989's aquatic horror quadruple feature, consisting of The Abyss, DeepStar Six, Leviathan and Lords Of The Deep. A more recent example is 2013, which saw two "Die Hard in the White House" movies go head to head, with Olympus Has Fallen and White House Down being released within months of each other. The latter was a big budget, PG-13 affair starring Channing Tatum and Jamie Foxx, and originally appeared to be the favorite to win that particular box-office battle.
Olympus Has Fallen starred Gerard Butler and Aaron Eckhart and thanks to a three month headstart on its rival, proved to be the winner. It didn't help that White House Down received bad reviews and lost out on its opening weekend to the likes of The Heat and World War Z. There wasn't any real talk of a White House Down 2 or potential franchise to follow the original movie leading up to its release, but had it been a hit, a follow-up almost certainly would have come together.
Of course, had a White House Down 2 happened, it's unlikely it would have featured Tatum or Foxx's characters under siege in the White House again. The later Olympus Has Fallen sequels found Butler's bodyguard protecting the President during an attack in London before the third movie found him on the run after being framed. A White House Down sequel likely would have taken a similiar approach, but while the movie has developed something of a cult in the years since its release, a follow-up is extremely unlikely to happen now, outside of a STV sequel.