Waivers are processed Wed, Thurs, Sat and Sun. The point is, if someone has the most free agent acquisition budget left, the other 11 owners have no way to stop it.
Example:
The owner of AP picked him up in the draft for $65. If that owner were to keep him the following season it would cost $75. The season is winding down and the AP team has the most free agent acquisition budget left. The next highest team has $39.
The owner could drop AP at the beginning of the week and immediate place a $40 claim on AP. No other team would have the budget to stop this from processing. The AP owner wouldn't even have to keep AP out of the lineup for a week to pull this off. Using the free agent acquisition price would allow that team to keep AP the next year for only $18, $57 cheaper than it should be.
This rule would only come into play if someone is gaming the system to decrease a player's keeper cost. If it is a legitimate FA acquisition, there shouldn't be any effect.