Born in Fort Huachuca, Arizona to a Puerto Rican father and a Korean mother, Robles got his start at the prestigious Tucson Soccer Academy, training under the guidance of coach Wolfgang Weber.

He was thrust into the starting role due to an injury to incumbent starter Tobias Sippel and outperformed fellow backup and German youth international Kevin Trapp.

Robles missed much of the summer training with Kaiserslautern – which had recently hired a new manager, Marco Kurz – because of his inclusion in the United States squad at the 2009 CONCACAF Gold Cup.

A few weeks later, Robles made contact with the league office and Ali Curtis who entered him in the Allocation draft, where he eventually signed with the Red Bulls.

[12] After first choice goalkeeper Ryan Meara suffered a hip injury, Robles was named the starter for the final four matches as well as the 2012 MLS Cup Playoffs.

For the second year in a row, Robles started all 34 league matches, and led his team to the 2014 Eastern Conference Finals where they ultimately fell to the New England Revolution 4–3 on aggregate.

[19] At the conclusion of the Red Bulls 2–1 loss to the Colorado Rapids on April 16, 2016, Robles became the MLS' new Ironman, by setting a new record of playing 90 minutes in 113 consecutive matches.

[20] During the Red Bulls next home match on April 25, previous record holder Kevin Hartman awarded Robles with the Ironman trophy.

[22] During a 6-game span in from May 13 to June 22, Robles helped the Red Bulls shut out opponents for 472 consecutive minutes; a streak that would come to an end in a 2–1 defeat to Real Salt Lake.

On March 6, Robles recorded a career high 13 saves to lead New York to a 2–0 victory in Mexico over Club Tijuana in the CONCACAF Champions League.

[33] Robles was named the 2019 New York Red Bulls Humanitarian of the Year for his work with children with serious illnesses, as well as his involvement with the Laureus Sport for Good fashion show.

In December 2019, Robles joined MLS expansion side Inter Miami CF ahead of their inaugural 2020 season.

[40][41] On July 11, 2009, Robles made his debut with the United States against Haiti in the 2009 CONCACAF Gold Cup, a game which ended drawn 2–2.