The first discovered extrasolar worlds -- giant, ``hot Jupiter'' planets orbiting remarkably close to their parent stars -- came as a surprise to solar-system-centric models of planet formation, prompting the development of new theories to produce these unexpected planets. The striking dearth of observed nearby planetary companions to hot Jupiters (~2%) has been widely quoted as evidence in support of high-eccentricity tidal migration: a framework in which hot Jupiters form further out in their natal protoplanetary disks before being thrown inward, stripping systems of any close-in planetary companions. By contrast, we demonstrate through a search for transit timing variations across the full four-year Kepler dataset that at minimum 13% of hot Jupiters have a nearby planetary companion. This subset of hot Jupiters must have a quiescent formation history such that they were able to retain nearby companions. We also demonstrate a ubiquity of nearby planetary companions to warm Jupiters, indicating that warm Jupiters typically form quiescently. We conclude with a new paradigm for the formation of short-period gas giants that is consistent with all current lines of evidence, in which >13% of hot Jupiters form in situ and the other <87% undergo high-eccentricity migration.