Bobbi began racing karts in 1995 and progressed to Formula Renault in 1998, where he recorded multiple podium finishes. In 2001 he competed in the Spanish Formula Euro Open Nissan Championship with Campos Racing, finishing sixth and taking one victory at Valencia. The following year he moved to Vergani Racing in the Formula Nissan 2000 series, winning six races and finishing as runner-up in the championship.
Bobbi tested a Formula One car for Minardi in 2000 and became the team's official test driver in 2001 — a position he retained for approximately three years. His manager throughout this period was Giovanni Minardi. The role included a Grand Prix weekend test session at the 2003 San Marino Grand Prix. Despite sustained involvement with the team across the 2002 and 2003 seasons, Bobbi never converted the test role into a World Championship race start. He also competed in the Rolex Sports Car Series with Cheever Racing during this period.
Alongside his Minardi duties, Bobbi developed a parallel sportscar career. In 2004 he raced for BMS Scuderia Italia in the FIA GT Championship, driving a Ferrari 550 GTS Maranello alongside Thomas Biagi across all eleven rounds. The pair won two races and finished third in the championship that year. In 2003 they had taken the title outright, also with BMS Scuderia Italia in a Ferrari 550, winning six races from ten rounds.
In 2006 Bobbi raced for AF Corse in the FIA GT Championship GT2 class, sharing a Ferrari F430 GT2 with Jaime Melo across nine of ten rounds. The pairing won two rounds and claimed the GT2 class championship.
Bobbi contested the 24 Hours of Le Mans in 2009 with Racing Box, driving a Lola B08/80-Judd in the LMP2 class alongside Andrea Piccini and Thomas Biagi. The car retired without being classified. He also raced in the Le Mans Series LMP2 class with Racing Box in 2009, winning one round and finishing ninth in the class standings, and made appearances in the Grand-Am Sports Car Series with Doran Racing in the same year.
In 2010 Bobbi raced in the FIA GT1 World Championship with both Triple H Team Hegersport in a Maserati MC12 and Marc VDS Racing in a Ford GT1 Matech across different rounds. He returned to the FIA GT1 World Championship in 2011 with DKR Engineering in a Chevrolet Corvette C6.R for two rounds. Later appearances included Blancpain Endurance Series entries with multiple teams in Ferrari 458 Italia GT3 machinery in 2011 and 2013, and outings in the Lamborghini Super Trofeo Europe in 2015.
After retiring from frontline competition, Bobbi joined Sky Italia as a Formula One analyst and commentator, working alongside fellow former driver Davide Valsecchi. In 2023, both were temporarily suspended by the broadcaster following sexist remarks made during an on-air post-race analysis of the Spanish Grand Prix.
Bobbi held FIA Platinum driver categorisation until 2013 and has been classified as FIA Gold from 2014 onward, reflecting his transition from the most demanding professional tier following his peak GT racing years.