16MHz is the processor speed. That means it can't react to anything in any way if it appears and disappears in less than a 16-millionth of a second.
Normally for SPI communication, you get a "clock" signal which says "look at the data line NOW" and then you look at the data pin to see if you are receiving a one or a zero. The key part is the "and then". That takes time. With a 16MHz processor, it will take two processor cycles to say "and" and "then". So that means you can't detect any incoming signal faster than 8MHz.
Now that's not entirely true - the Arduino chip has specialised hardware which does the SPI without bothering the main processor. But 8MHz is as fast as it will go. 4MHz is the default transmission speed.
It may work, the receiver may be able to decode 13MHz, but I don't expect that it will work reliably.