16-to-4 Encoder Using 8-input NAND
Yeah, sure.  One condition though - I'm assuming all 16 input lines are normally high with only one going low.   I'm going to ignore cases of multiple lines being low, or none being low.

Note that any bit of the output is going to be high for half of the input states, and low for the other half.  Half of 16 is 8.   

Imagine connected the 8 inputs of a gate to the first 8 of the 16 input lines.  For those eight input states where one of those lines is low, the logical "and" fails, and the NAND puts out high.  For the other eight input states, the "and" is true so the NAND puts out a low.   Looks like we found the MSB (most significant bit) of a four-bit binary number.

The other bits work similarly. Their gates need to be wired up differently.