A $d$-dimensional bar-and-joint framework $(G,p)$ with underlying graph $G$ is called universally rigid if all realizations of $G$ with the same edge lengths, in all dimensions, are congruent to $(G,p)$. A graph $G$ is said to be generically universally rigid in $\mathbb{R}^d$ if every $d$-dimensional generic framework $(G,p)$ is universally rigid. In this paper we focus on the case $d=1$. We give counterexamples to a conjectured characterization of generically universally rigid graphs from R. Connelly (2011). We also introduce two new operations that preserve the universal rigidity of generic frameworks, and the property of being not universally rigid, respectively. One of these operations is used in the analysis of one of our examples, while the other operation is applied to obtain a lower bound on the size of generically universally rigid graphs. This bound gives a partial answer to a question from T. Jordán and V-H. Nguyen (2015).