Yes it is very possible. Especially if the bully stick (and rawhide chews) are left outside, and remain damp (or wet from rain). The amount of bacteria growth goes off the charts.
A dear friend almost lost her two Wire Fox Terriers. She would leave them during the day with big rawhide bones in their outside run. They became deathly ill over a period of time. Everyone was trying to figure out what was making them so very, very sick. Finally, her vet asked about their daily routine, and it was discovered. She brought in the rawhide and he took a scrapping of the rawhide and looked at it under the microscope. Ah ha!! All turned out fine, but after dealing with many trips to the vet and a lot of money spent.