3. Chris Bosh
Year one of the James Harden-Dwight Howard pairing saw the Rockets improve from the eighth seed in the Western Conference to the fourth seed. However, the Rockets were exposed in the playoffs when they faced the Portland Trail Blazers, who had a pairing of their own, with Damian Lillard and LaMarcus Aldridge, who proved to be too much for the Rockets, as they combined for 55.3 points per game in the opening round series.
The Rockets were top-heavy and needed a third star, as they were in the Big Three era. The Miami Heat gad just reached the NBA Finals four years in a row with a trio of LeBron James, Dwyane Wade, and Chris Bosh and the Oklahoma City Thunder reached the Finals just two seasons prior with a Big Three of Kevin Durant, Russell Westbrook, and James Harden.
The Rockets made a full run at Chris Bosh, who was fresh off four consecutive All-Star appearances and two championships in the previous three seasons. Nabbing Bosh, who was capable of being an offensive force (16.6 points on 51.2 percent shooting as the third wheel in Miami) would've done wonders for the Rockets, especially considering the injury woes to Dwight Howard prior to him joining the Rockets (which surfaced in the following season).
Ultimately Bosh rejected the Rockets' overtures and opted to remain in Miami on a five year deal worth $118 million. Bosh would later explain that although the Rockets would've formed a lethal trio with him, Harden, and Howard, the pressure of winning with three superstars made it unappealing,
"They were trying to win right away. And I was really happy to be touted that I possibly could've been out there. But you know, that doesn't guarantee anything, and I know that. All that guarantees is a bunch of pressure."
The Rockets were certainly left hanging at the altar, as they were under the belief that they had a commitment from the Dallas native.