I had 15 until a few minutes ago when I gave some away.
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It seems that Niantic made a change last night and made gifts in your bag cap out at 5. Earlier in the day yesterday, myself and several other people had 10+ gifts in our bags, but after sending them out and restocking, I've yet to see anyone able to collect more than five.
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Good to know.
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Apparently, Niantic made another tweak. It appears the new max is 10