We consider the following fragmentation model with cutoff: a fragment with initial size $x_{0}>1$ splits into $b>1$ daughter fragments with random sizes, the partition law of which has exchangeable distribution. In subsequent steps, fragmentation proceeds independently for each sub-fragments whose sizes are bigger than some cutoff value $x_{c}=1$ only. This process naturally terminates with probability $1$. The size of a fragment is the random mass attached to a leaf of a ''typical'' path of the full (finite) fragmentation tree. The height's law of typical paths is first analyzed, using analytic and renewal processes techniques. We then compute fragments' size limiting distribution ($x_{0}\uparrow \infty $), for various senses of a typical path. Next, we exhibit some of its statistical features, essentially in the case of the exchangeable Dirichlet partition model.
