Yes, it is possible to get both the Mach Bike and Acro Bike at the same time.
First, you need to show the Mach Bike to the Hex Maniac in the desert of Route 111.
Next, you need to show the Acro Bike to a Bird Keeper in the area that requires the Acro Bike at Route 119, you will need Surf and Waterfall.
Finally, you need to speak to the triathlete in the battle resort with any bike.
After you're done doing this, go to Rydel's Cycles and talk to him, he should give you the bike that you don't currently have.
Hope this helps!