While in fantasy leagues teams drop to the back of the pack after getting one waiver claim, that is not how it works in the actual NFL.
When a team is listed #1 in the waiver claim order, they can claim as many players as they want for as long as they hold that #1 spot and get all of those players. That's why in the "mini-draft" immediately following cut day at the end of the preseason, teams who were early in the draft order often get multiple players.