Can I restack the buckets?
Jelly, 6 5 bytes

Thanks to @Lynn for saving 1 byte.

ṢFµJ⁼


Try it online! (comes with test-suite footer)

Explanation

ṢFµJ⁼    Main link. Argument: piles
Ṣ          Sort the piles by the size of the top bucket.
 F         Stack the piles, putting the left one to the top.
   J       See what a full pile with this many buckets would look like.
    ⁼      See if that looks like the pile you built.