Can the number be split into powers of 2?
05AB1E, 9 8 bytes
Ýos.œåPZ

-1 byte thanks to @Emigna by using Z (max) for the list of 0s and 1s to mimic an any command for 1 (truthy).
Try it online or verify all test cases. (NOTE: The т in the header is 100 to only get the first 100 power of 2 numbers, instead of the first input amount of power of 2 numbers. It works with the input amount of power of 2 as well, but is pretty inefficient and might timeout on TIO if the input is large enough.)
Explanation:

Ý            # Create a list in the range [0,n], where n is the (implicit) input
             # (or 100 in the TIO)
             #  i.e. 81024 → [0,1,2,3,...,81024]
 o           # Raise 2 to the `i`'th power for each `i` in the list
             #  → [1,2,4,8,...,451..216 (nr with 24391 digits)]
  s          # Swap to take the input
   .œ        # Create each possible partition of this input
             #  i.e. 81024 → [[&quot;8&quot;,&quot;1&quot;,&quot;0&quot;,&quot;2&quot;,&quot;4&quot;],[&quot;8&quot;,&quot;1&quot;,&quot;0&quot;,&quot;24&quot;],...,[&quot;8102&quot;,&quot;4&quot;],[&quot;81024&quot;]]
     å       # Check for each if it's in the list of powers of 2
             #  → [[1,1,0,1,1],[1,1,0,0],...,[0,1],[0]]
      P      # Check for each inner list whether all are truthy
             #  → [0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,0]
       Z     # Take the maximum (and output implicitly)
             #  → 1 (truthy)