I have made a lot of vases, bowls, mugs and pitchers and never had a problem of them "falling apart". I have used all kinds of wood including red oak and all thicknesses of wood.
I always used Titebond II wood glue except for the mugs and pitchers that may have liquid put in them and then I use Titebond III for the water proof feature of that glue.
One thing that might help if you are making bowls is a "bowl press" for gluing. Please find a copy of "Wooden Bowls from the Scroll Saw" written by Carole Rothman. It not only shows you how to make the press, it has a BUNCH of designs and ideas for you to use. If your local library does not have a copy of this book, it can be purchased from Fox Chapel Publishing for $19.95 plus shipping. (anyway that is what I paid on 10/2/2009 for a total of $24.22.) and is worth twice that price.
Anyway have FUN and make some beautiful vases and bowls.
Rog