There appears to be a hard limit to the number of gifts you can open (20 a day?). The limit of 5 unsent gifts at a time in the bag can be avoided by sending gifts as soon as spinning a stop yields one (so you never even reach the 5-in-a-bag limit). But there seems to be a limit to the number of gifts that spinning stops will yield in a day, though I still don’t know what that is.
Did everybody notice after a day or so of the friend feature’s rollout that the number of items you get when opening a gift dropped off significantly? Initially a typical gift would have (for example) 5 ultra balls, 5 pineaps, and a revive, or even five great balls and 3 raspberries; but starting yesterday I noticed the 5-at-a-time dropped to 1.
It makes sense to me, since the original gifts would fill up your bag with just 30 friends at a time, when factoring in all the items given by the pokestop spins when collecting gifts to send, plus the daily/weekly bonuses.