Unless all inputs incur precision problems, Qhull will terminate. Qhull adjusts the inner and outer planes to account for the joggled input. If a precision problem occurs, Qhull merges one of the offending facets into one of its neighbors. Since all precision problems in Qhull are associated with one or more facets, Qhull will either terminate or attempt to merge the last remaining facets. Neither method has an upper bound for the width of the output facets, but both methods work well in practice.
Joggled input is easier to justify.
Precision errors occur when the points are nearly singular. For example, four points may be coplanar or three points may be collinear. Consider a line and an incident point. A precision error occurs if the point is within some epsilon of the line. Now joggle the point away from the line by a small, uniformly distributed, random quantity.