Yes, there are some spots in the game, including the Safari Zone, which you need both bikes.
1. Talk to a Hex Maniac in the Route 111 Desert and show her the Mach Bike.
2. Talk to a Bird Keeper in Route 119 and show him the Acro Bike. (Also requires Surf and Waterfall)
3. Then talk to a Triathlete in the Battle Resort, then you can go back to Rydel to get the other bike.
Due to step 3, you cannot have both bikes at once until you beat the E4 and the Delta Episode.