It depends on a lot of things (climate, specific personal requirements, etc). My wife & I took our two young children -- 2 and 4 -- to California & Hawaii this summer without checking any bags. It was a 17 day trip. This is way more challenging on business trips when you usually need to pack nicer clothing.
"Checked luggage is lost luggage."
I survived a month in Europe shortly after with carry-ons. It was doable, pleasant even, but I had to make some sacrifices: one pair of shoes, no laptop, no heavy jacket. It's obviously not ideal and possibly not feasible for someone traveling with a non-ticketed baby to travel this way.