In this paper, a very tiny HW design of a Quantized Fully Convolutional Neural Network is proposed which demonstrates that accurate Human Posture Recognition can be designed by exploiting only pressure sensors and keeping the computation close to the data sources, according to the edge computing paradigm. The custom design of the QFCN exploits a base-2 quantization scheme to achieve state-of-the-art performances in terms of classification accuracy, together with a very reduced number of mapped physical resources and power consumption. Trained and validated on a public dataset for in-bed posture classification, the QFCN exhibits an accuracy up to 96.77% in recognizing 17 different postures. When prototyped on a Xilinx Artix 7 FPGA the design achieves less than 7 mW dynamic power dissipation and a maximum operation frequency of 26.6 MHz, compatible with an Output Data Rate (ODR) of the sensors of 9.13 kHz.