Answering my own question: $500.00 worked, $1,000.00 didn't (I saw the failure message on the CSR's screen). The CSR said the limit was $1,000, but since that didn't work, either the limit is actually $999.99 (i.e. the limit is "$1,000 or more" and not "more than $1,000") or the CSR was mistaken. I will continue to experiment.

Another possibility is, the Kroger GC I was reloading had a few cents on it; is there a maximum value for Kroger GCs? Would it happen to be $1,000?