  In this note, we show that there exist cusped hyperbolic $3$-manifolds that
embed geodesically, but cannot bound geometrically. Thus, being a geometric
boundary is a non-trivial property for such manifolds. Our result complements
the work by Long and Reid on geometric boundaries of compact hyperbolic
$4$-manifolds, and by Kolpakov, Reid and Slavich on embedding arithmetic
hyperbolic manifolds.
