Most leauges have a cutoff date of eligibility. For example, Major League Baseball players must be on their team's roster by September 1st in order to be considered for the playoffs. Players added after that date would only be eligible in case of a rash of injuries, which of course no one would wish for.
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Yes they can. If the player was not on a roster for week 17 or waived by the Thursday before Week 17. They can be signed by a team during the playoffs.
The free agent signing period begins in March. Under the new collective bargaining agreement, during the offseason teams will be able to expand their rosters to 90 players.