  We study Fano manifolds $X$ admitting an unsplit dominating family of
rational curves and we prove that the Generalized Mukai Conjecture holds if $X$
has pseudoindex $i_X = (\dim X)/3$ or dimension $\dim X=6$. We also show that
this conjecture is true for all Fano manifolds with $i_X > (\dim X)/3$.
