Falls have serious consequences and are prevalent in acute hospitals and nursing homes caring for older people. Most falls occur in bedrooms and near the bed. Technological interventions to mitigate the risk of falling aim to automatically monitor bed-exit events and subsequently alert healthcare personnel to provide timely supervisions. We observe that frequency-domain information related to patient activities exist predominantly in very low frequencies. Therefore, we recognise the potential to employ a low resolution acceleration sensing modality in contrast to powering and sensing with a conventional MEMS (Micro Electro Mechanical System) accelerometer. Consequently, we investigate a batteryless sensing modality with low cost wirelessly powered Radio Frequency Identification (RFID) technology with the potential for convenient integration into clothing, such as hospital gowns. We design and build a passive accelerometer-based RFID sensor embodiment---ID-Sensor---for our study. The sensor design allows deriving ultra low resolution acceleration data from the rate of change of unique RFID tag identifiers in accordance with the movement of a patient's upper body. We investigate two convolutional neural network architectures for learning from raw RFID-only data streams and compare performance with a traditional shallow classifier with engineered features. We evaluate performance with 23 hospitalized older patients. We demonstrate, for the first time and to the best of knowledge, that: i) the low resolution acceleration data embedded in the RF powered ID-Sensor data stream can provide a practicable method for activity recognition; and ii) highly discriminative features can be efficiently learned from the raw RFID-only data stream using a fully convolutional network architecture.