As I understand it, your question has the answer no.
Since you ask for $1-$designs, $\lambda$ is essentially how many times one of the $v$-many points appear in a block, which has size $k=4$ in the design $D$. Start with a design $D'$ on $16$ points. I arrange them in a square and choose for blocks the rows, columns, and both (extended) diagonals, giving $16$ blocks and $\lambda=4$. Now multiply this design by 3 to get $D$ on $v=48$ points with the desired parameters. Any partition into sets of size $3$ has to have two or more sets "cross" different copies of $D'$. One can modify this to get larger "clumps", but if your $D$ falls into two or more pieces on $v'$ and $v''$ points where one of them is not a multiple of $3$, then you cannot refine that design into a partition of $3-$sets as you desire.
I have not verified it, but I suspect that this can be modified to a "connected" example where one still fails to refine such a design into a partition into $3-$sets.
Gerhard "I Think That Covers It" Paseman, 2013.12.17