Following Week 3 of the regular season, the NFL's waiver wire priority system resets. No longer is the priority determined by last year's record; it is now based off of how teams have performed thus far in 2013.
Prior to Week 3, the Patriots had the 29th priority on the waiver wire, which is to say that if a player was waived, there were 28 teams ahead of them who could claim him and have priority of being awarded him ahead of the Patriots.
Now that Week 3 has passed, the order has reset, which it will continue to do each week of the season.
The Buccaneers, by virtue of their 0-3 record and strength of schedule, now have the top priority in the waiver system. To break a tie between teams with the same record, the record of opponents already played is tabulated. If two teams have the same record, whichever has played the superior opponents (based on won-loss record) will have priority. If two teams have the same record and same won-loss record among opponents, a coin-flip will be used to break a tie.
The Patriots are one of seven teams with a 3-0 record, though the Seahawks, Saints and Bears have an opponent's combined record of just 2-7, while the Patriots' opponents have a combined record of 3-6. Because of that, the Patriots have priority over those teams if they were to both put in a claim.
If the Patriots were to claim the same player as either the Chiefs or Broncos -- who have also played opponents with a combined 3-6 record -- the league would use a coin flip to determine which team the player was awarded to.
If the Patriots were to claim the same player as either the Dolphins, Miami would be awarded said player because they have played a more difficult schedule (four wins by their three opponents).
So, as things currently stand, the Patriots have the 27th priority on the waiver wire, shared by the Chiefs and Broncos.