A partial Steiner triple system is is $sequenceable$ if the points can be sequenced so that no proper segment can be partitioned into blocks. We show that, if $0 \leq a \leq (n-1)/3$, then there exists a nonsequenceable PSTS$(n)$ of size $\frac{1}{3}\binom{n}{2}-a$, for all $n \equiv 1 \pmod{6}$ except for $n=7$.