In many books on Kobayashi hyperbolic spaces (eg Lang's and Kobayashi's), various conditions are discussed that imply completeness. I assume there must be an example of a complex manifold that is Kobayashi hyperbolic but such that the Kobayashi metric is not complete. What is such an example?
I thought about finite-type Riemann surfaces a bit. I don't think there are non-complete Kobayashi hyperbolic spaces among these. If the Riemann surface is uniformized by either the Riemann sphere or the complex plane, then the surface is not Kobayashi hyperbolic. If the surface is uniformized by the unit disc, then the Kobayashi metric is the hyperbolic metric, and in this case the metric is complete.
EDIT: After more reading in Kobayashi's book, I found a simple example. If you take a Kobayashi hyperbolic space X that contains a complex submanifold A of codimension at least 2, then X-A is Kobayashi hyperbolic but not complete.
Both properties follow from the fact $d_{X-A} = d_{X} |_{X-A}$, and hence the "A boundary" is only finite distance away. If we think of this equality as two inequalities, one follows directly from the distance non-increasing property of the Kobayashi metric under holomorphic maps. The other inequality is proved by taking the maps from the disk that you use to define the Kobayashi metric and perturbing them a little so as to miss A (which needs that A has codim at least 2).