Can a bulging disc pinch more than one nerve at a time?

Bulging disc. It can depending on the position. It can effect the more lateral exiting nerve as well as the more central transcending root the exits at the next level below.
Absolutely. When a disk herniation is large enough, it can not only pinch one or two nerves, it can cause severe compression of the sac of nerves pinching the entire sac and many nerves.