The fundamental sensing limit of pinching-antenna systems (PASS) is studied from a Bayesian Cramér-Rao bound (BCRB) perspective. Compared to conventional CRB, BCRB is independent of the exact values of sensing parameters and is not restricted by the unbiasedness of the estimator, thus offering a practical and comprehensive lower bound for evaluating sensing performance. A system where multiple targets transmit uplink pilots to a single-waveguide PASS under a time-division multiple access (TDMA) scheme is analyzed. For the single-target scenario, our analysis reveals a unique mismatch between the sensing centroid (i.e., the optimal PA position) and the distribution centroid (i.e., the center of the target's prior distribution), underscoring the necessity of dynamic PA repositioning. For the multi-target scenario, two target scheduling protocols are proposed: 1) pinch switching (PS), which performs separate pinching beamforming for each time slot, and 2) pinch multiplexing (PM), which applies a single beamforming configuration across all slots. Based on these protocols, both the total power minimization problem under a BCRB threshold and the min-max BCRB problem under a total power constraint are formulated. By leveraging Karush-Kuhn-Tucker (KKT) conditions, these problems are equivalently converted into a search over PA positions and solved using an element-wise algorithm. Numerical results show that i)~PASS, endowed with large-scale reconfigurability, can significantly enhance the sensing performance compared with conventional fixed-position arrays, and ii)~PS provides more robust performances than PM at the cost of higher computational complexity.