This has been educational.   
I ran a re-scan.  The scan found 518 blocks missing from my client's wallet.  The re-scan didn't add the blocks to the wallet, which is normal if the blocks aren't for an address found in the wallet.  However....
I backed up the wallet and created a new one, then imported the private keys from the older wallet into the new one.  I also downloaded the blockchain from scratch.  After that I did another re-scan.  That scan found "only" 167 blocks missing from the wallet just created.  The blockchain explorer is returning a 500 error at the moment, so looking up a few of them will be a chore for another day.  Here are a few of them from the end of the list found by the scan:

AddToWallet a7f2f06bb5
AddToWallet ce3fae5d6d
AddToWallet 5c6facb254

Here's a few from the beginning of the list:

AddToWallet 09230627f5
AddToWallet ac1c549ea2
AddToWallet 028c82e01b

I have some (a smaller number) of missing blocks from the Qt wallets of other coin clients based on bitcoin-qt, so this does not look like anything specific to the Nanotoken Qt or CLI clients.