SPI with a level converter in the other end of connection
More than probably your level shifter is not fast enough.
A level shifter uses a mosfet to copy signals from one voltage level to the other, but each mosfet has a latency or, if you prefer, a rise time.
Your board probably doesn't use a fast switching mosfet and so the SPI protocol (which is a quite fast one) is not copied correctly.

On top of that you actually don't need a level switcher for the nRF24 modules: the power pin must be limited to 3.3V, but the signal pins are 5V tolerant. 
This means you only need a 3.3V voltage source and all the remaining connections can go directly (and safely) into your Arduino board.

UPDATE: the speed concept still applies, it doesn't matter which side of the connection you are using. A mosfet has a little capacitance which slows down the signal transmission: higher the capacitance, slower the maximum data rate.
The cheap chinese level shifter is not designed for high speed communication, while the Texas Instrument device is clearly designed for high speed.
If you look at the SN74HCT245 datasheet you'll notice the switching speed is reported for 50pF capacitance and ranges from 8ns to 16ns: that's the time needed to build enough charge in the switching circuit.