Hi, thanks. 
I will see if I can give the Pi transmitter a better power supply, and reduce the transmit power level. I notice that when the Arduino is (attempting) to receive from the Raspberry Pi the carrier LED does not come on as brightly as it does when it is successfully receiving from the other Arduino. I did try a cap across Vcc-GND on the Pi but it made no difference. So, yeah - I might try an external power supply for the radio tied to the same ground.
I have tried several modules on it in both transmitter and receiver mode. I have three different modules from three different sources (all NRF24L01+). They all receive, although, interestingly, the really cheap one ($4) is 100% reliable as a receiver, whereas the two somewhat more expensive ones I tried got some garbled packets.
