You'd have to get the other top picks to accept whatever the lower offer was. So you could sign say a bunch of college seniors for 20K a pop.
If a player from the draft pool rounds doesn't sign, that money is removed from the pool.
There is also nothing stopping a team from picking him and trying to get him to change his mind. If he doesn't sign they get a pick in next year's draft. Not ideal but it does make it less risky.