MCP73871 LiPo low charge current
Some improvements and checks you should make no matter what


Add an input capacitor of at least 10 uF.
Add an output (load) capacitor of at least 4.7 uF.
Check the voltage rating and characteristic of all capacitors (including those already included in your PCB). The capacitance of certain ceramic capacitors can drop down to just 10% of its nominal value when applying as little as 3V.
Add an LED + resistor at the /PG output. It may help with debugging/troubleshooting.
Check all solder joints.
Check all wires and or PCB tracks. Make sure that they can withstand the currents that will flow through them. Also, make them as short as possible.
Add a heatsink, either on top of the chip or by properly bonding the thermal pad. If in doubt, do both at least during the troubleshooting phase.
Check the values of R16 and R17 as they may be wrong or swapped. Desolder them if needed for measuring.
Check the thermistor value. In your schematic it is 15k, whereas it should be a 10k NTC.
Check the battery ratings. Is it 3.7V/4.2V? Can it accept the charging/discharging current you pretend to use?
Use a better (lower AWG) charging cable (or wires) to avoid voltage drops in your bus (input) voltage.
Check that \$V_{batsense}\$ is properly connected. 


Who are the main suspects?


R16: it may actually be 10k instead of 1k (thus limiting charging to about 100 mA). If \$V_{bat}\$ is well above 3.0V after the battery has been charging for a while at 100 mA, that may be the case.
Thermal regulation kicking in: a poor thermal bonding/heatsinking of the IC may be triggering the thermal regulation mechanism and throttling down the charging current. May have the same symptoms as above, but with a correct R16 value.
Bad battery: the battery may be faulty or may not be up to your needs. If \$V_{bat}\$ is well below 3.0V after the battery has been charging for a while at 100 mA, that may be the case. You could be stuck at the preconditioning state (10% of charge current) and not be able to switch to the full charge state.
Bad \$V_{batsense}\$ connection: may have the same symptoms as above, even with a perfectly fine battery. 


Also, always have at hand these tables from the datasheet when troubleshooting: