Pressure measurement using a Pressure Sensitive Mat (PSM) enables a non-contact approach for monitoring patient vital signs such as respiration rate (RR) and heart rate (HR). Non-contact patient monitoring in Neonatal Intensive Care Units (NICU) can improve the quality of patient care and reduce patient discomfort. A clinical trial was conducted with the Children's Hospital of Eastern Ontario (CHEO) collecting PSM data from 15 patients in three different bed types. Algorithms for estimating RR were developed. The best RR estimation algorithm worked well when the patient is on ventilator support achieving a mean absolute error (MAE) of 3.92 +/- 6.61 bpm on clean data without patient movements or interventions. When the RR estimation algorithm is evaluated on all patients, RR estimation performance is mixed. In particular, very low mass neonates and patients laying on multiple blankets presented challenges for RR estimation.