I've made about a dozen bows 1 1/4" wide from 3" to 3 1/2" Madake bamboo. 3" is really pushing it, but if the wall thickness of the culm is good(over 1/4"), it works. 3 1/2" pole will yield 4 splits easily.
4" is better, but good luck finding poles that size of a specie strong enough for bow backing.