When our family got bigger (11 kids), my sister (who was purchasing a house and pregnant with baby #3) called a siblings reunion to talk about this. She explained that her finances didn't allow her to spend her family's money on all her nieces and nephews anymore. The meeting came at the right time. Another sister had just lost her job and our brother was also expecting a new baby.
We agreed that from now on, only the godparents would offer gifts for Christmas. [in our family, aunts and uncles are the godparents] This way the financial burden was lessen for everybody and equally distributed. Each kid gets less gifts but the gifts are of higher value (giving for example 3x$25 instead of 11x$10).
The kids still get plenty of gifts from grand-parents, great-grand-parents, friends, and even some great aunts and uncles and corporations (they live in France where many company/unions offer Christmas gifts for kids)
For the birthday, only the birthday kid get a present, and only if there is a party/celebration of some sort.
And, this will last at least until they turn 18.