How to run a bash command until it is successful or a timeout is reached?

You can simplify things a bit by putting command right in the test and doing increments a bit differently. Otherwise the script looks fine:

NEXT_WAIT_TIME=0
until (( NEXT_WAIT_TIME == 5 )) || cmd; do
    sleep "$(( NEXT_WAIT_TIME++ ))"
done
(( NEXT_WAIT_TIME < 5 ))