  Let $T$ be a satellite knot, link, or spatial graph in a 3-manifold $M$ that
is either $S^3$ or a lens space. Let $b_0$ and $b_1$ denote genus 0 and genus 1
bridge number, respectively. Suppose that $T$ has a companion knot $K$ and
wrapping number $\omega$ with respect to $K$. When $K$ is not a torus knot, we
show that $b_1(T)\geq \omega b_1(K)$. There are previously known
counter-examples if $K$ is a torus knot. Along the way, we generalize and give
a new proof of Schubert's result that $b_0(T) \geq \omega b_0(K)$. We also
prove versions of the theorem applicable to when $T$ is a ``lensed satellite''
or when there is a torus separating components of $T$.
