Pinching antenna system (PASS) has recently shown its promising ability to flexibly reconfigure wireless channels via dynamically adjusting the positions of pinching antennas over a dielectric waveguide, termed as pinching beamforming. This paper studies the fundamental limit of the sum rate for a PASS-assisted multiple access channel, where multiple users transmit individual messages to a base station under the average power constraint. To this end, a dynamic pinching beamforming setup is conceived, where multiple pinching beamforming vectors are employed in a transmission period and the capacity-achieving non-orthogonal multiple access (NOMA) based scheme is considered. For the ideal case with an asymptotically large number of pinching beamforming vectors, the optimal transmission scheme is unveiled to carry out alternating transmission among each user whose channel power gain is maximized with the tailored pinching beamforming. This implies that NOMA is not needed for achieving the sum capacity and the required optimal number of pinching beamforming vectors is equal to the number of users. With this insight, the corresponding sum rate is derived in closed-form expression, which serves as the upper bound of the sum rate. Inspired by this result, a lower bound of the sum rate under an arbitrarily finite number of pinching beamforming vectors is obtained. Numerical results validate our theoretical findings and also illustrate the practical significance of using dynamic pinching beamforming to improve the sum rate.