A collection C of balls in R^d is \delta-inflatable if it is isometric to the intersection U \cap E of some d-dimensional affine subspace E with a collection U of (d+\delta)-dimensional balls that are disjoint and have equal radius. We give a quadratic-time algorithm to recognize 1-inflatable collections of balls in any fixed dimension, and show that recognizing \delta-inflatable collections of d-dimensional balls is NP-hard for \delta \geq 2 and d \geq 3 if the balls' centers and radii are given by numbers of the form a+b\sqrt{c+d\sqrt{e}}, where a, ..., e are integers.