The photon detection system for the LHCb RICH Upgrade consists of an array of multianode photomultiplier tubes (MaPMTs) read out by custom-built modular electronics. The behaviour of the whole chain was studied at CERN using a pulsed laser. Threshold scans were performed in order to study the MaPMT pulse-height spectra at high event rates and different photon intensities. The results show a reduction in photon detection efficiency at 900 V bias voltage, marked by a 20% decrease in the single-photon peak height, when increasing the event rate from 100 kHz to 20 MHz. This reduction was not observed at 1000 V bias voltage.