For the Princess Sirindhorn Neutron Monitor in Thailand, we find that the cross-counter leader fraction (XLF), i.e., the inverse of multiplicity across counters, depends on counter separation and differs for end and middle counter tubes. Multiplicity at small separation can be attributed to single secondary particles, while multiplicity at large separation indicates multiple secondaries from the same cosmic ray shower. The end/middle differences are clarified using follow-up measurements of 1) analog neutron signals from selected counters and 2) digital timing data from all counters for events of interest. Considering Counters 17 and 18 at the edge of the monitor, triggering on either counter leads to the same event rate on the other, so small-separation XLF depends on the count rate of the first (triggering) counter, which is lower for an end counter. To examine large-separation multiplicity, due to multiple secondaries, an FPGA-based readout system was triggered by Counter 2 followed by Counter 18 within 250 microseconds, while also monitoring Counter 10 in between. Counter 10 exhibited an enhanced rate, indicating a few events that densely “carpeted” the neutron monitor, but the majority of triggers involved a sparse distribution of isolated secondary particles. This is consistent with the digital timing data from all counters and end/middle differences in XLF.