We measured count rates and scatter fraction on the Discovery STE PET/CT scanner in conventional 2D and 3D acquisition modes, and in a partial collimation mode between 2D and 3D. As part of the evaluation of using partial collimation, we estimated global count rates using a scanner model that combined computer simulations with an empirical live-time function. Our measurements followed the NEMA NU2 count rate and scatter-fraction protocol to obtain true, scattered and random coincidence events, from which noise equivalent count (NEC) rates were calculated. The effect of patient size was considered by using 27 cm and 35 cm diameter phantoms, in addition to the standard 20 cm diameter cylindrical count-rate phantom. Using the scanner model, we evaluated two partial collimation cases: removing half of the septa (2.5D) and removing two-thirds of the septa (2.7D). Based on predictions of the model, a 2.7D collimator was constructed. Count rates and scatter fractions were then measured in 2D, 2.7D and 3D. The scanner model predicted relative NEC variation with activity, as confirmed by measurements. The measured 2.7D NEC was equal or greater than 3D NEC for all activity levels in the 27 cm and 35 cm phantoms. In the 20 cm phantom, 3D NEC was somewhat higher (~15%) than 2.7D NEC at 100 MBq. For all higher activity concentrations, 2.7D NEC was greater and peaked 26% above the 3D peak NEC. The peak NEC in 2.7D mode occurred at ~425 MBq, and was 26-50% greater than the peak 3D NEC, depending on object size. NEC in 2D was considerably lower, except at relatively high activity concentrations. Partial collimation shows promise for improved noise equivalent count rates in clinical imaging without altering other detector parameters.