According to the GYSA President's Cup website FAQ section, it does appear that one can play in both President's and State Cup in the same year. However, full teams cannot participate in both tournaments.
http://www.soccerincollege.com/TTMore.aspx?tid=PRESGA&tab=4&Qual=MISC&Seq=200
"7. Can a player play in both the President’s Cup and State Cup?
Yes, a player may play in the President’s Cup and then can transfer, within GYSA rules, to a team that is playing in the State Cup."
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