The uplink data rate region for interfering transmissions in wireless networks has been characterised and proven, yet its underlying model assumes a complete temporal overlap. Practical unplanned networks, however, adopt packetized transmissions and eschew tight inter-network coordination, resulting in packet collisions that often partially overlap, but rarely ever completely overlap. In this work, we report a new design called Partial Symbol Recovery (PSR), that specifically targets the parts of data symbols that experience no interference during a packet collision. PSR bootstraps a successive interference cancellation (SIC) like decoder from these strong signals, thus improving performance over techniques oblivious to such partial packet overlaps. We have implemented PSR on the WARP software-defined radio platform and in trace-based simulation. Our performance evaluation presents experimental results from this implementation operating in a 12-node software network testbed, spread over two rooms in a non-line-of-sight indoor office environment. Experimental results confirm that our proposal PSR decoder is capable of decoding up to 60% of collided frames depending on the type of data and modulation used. This consistently leads to throughput enhancement over conventional Wi-Fi under different scenarios and for the various data types tested, namely downlink bulk TCP, downlink videoon-demand, and uplink UDP.