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According to Sony's official channels, most of those problems have been solved as of this afternoon.
The first issue was customers having issues just signing into PSN:
The second issue was gamers trying to redeem their PlayStation Plus trial vouchers:
Finally, gamers were also having issues accessing the PSN store:
The first issue was customers having issues just signing into PSN:
PS4 users: If you had trouble signing in to PSN earlier, try again. The network issues should be mostly resolved now.
-- Ask PlayStation (@AskPlayStation) November 16, 2013
The second issue was gamers trying to redeem their PlayStation Plus trial vouchers:
PS4 users: If you were having trouble redeeming your PS Plus trial earlier, please try again now.
-- Ask PlayStation (@AskPlayStation) November 16, 2013
Finally, gamers were also having issues accessing the PSN store:
Having trouble accessing the PSN Store from your PS4 today? The Store is actually up, so try restarting the console.
-- Ask PlayStation (@AskPlayStation) November 16, 2013